Zak Kaczmarek/Getty Images 1.She is fighting the pernicious idea that older women don’t matter. “People say I’m controversial,” Madonna told an audience of music-industry peers in 2016. “But I think the most controversial thing I have ever done is to stick around.” Sexism was the demon that haunted Madonna’s early career, but for two decades — maybe longer — it’s had an equally unwelcome sibling: ageism. Madonna was a pioneer of welding her voice to her image, and in a culture consumed with critiquing how women look, and controlling how they use their bodies, she’s been on the front lines — a seductress and a battering ram. But as she’s continued to be a force while she deigns to grow older, she’s faced a new frontier of abuse. There has never been a pop star writing and performing at her level, and demanding a seat at the table, at her age. Why wouldn’t Madonna demand it? One of the great conundrums of the internet era is pop culture’s short memory; the receipts are right there to be found, yet few bother to do the looking. But yes, in a career spanning four decades, Madonna made real cultural change, and caused a few cultural crises, over and over again. For all the criticism she’s weathered during four decades in the spotlight, she deserves a celebration. —Caryn Ganz

Orion Pictures, via Photofest 2.She turned her confidence and style into movie stardom. The first time we see Madonna in “Desperately Seeking Susan,” she’s lying on the floor of an opulent, disarrayed Atlantic City hotel room, taking Polaroids of herself. (Of course she was early to selfies.) The movie, an unlikely box-office and critical hit in 1985, was her first major screen role: She plays the titular hyper-charismatic grifter, a downtown Holly Golightly with actual sex appeal, waaaay better jewelry and one badass jacket. The cropped Santo Loquasto tux with the gold pyramid on the back became a cinematic icon (in 2014, it sold at auction for over $250,000), but the whole movie is essentially a search for the perfect accessory. There are stolen Nefertiti earrings, and the spike-heeled studded boots that Madonna picks up at the real vintage emporium Love Saves the Day. That accumulation of style points as a means to self-expression is quintessential early Madonna. Susan Seidelman, the director, knew Madonna through downtown circles and pitched her for the part. She wanted to explore identity and reinvention, with an authentically gritty New York as the canvas. Madonna studied acting for the role, but it hardly mattered — her character’s audacious cool and street-savvy hustle is all her own. When her second album hit during production, she suddenly became a megastar. The audience got the message: Being who you want pays off. —Melena Ryzik

Jim Steinfeldt/Michael Ochs Archives, via Getty Images 3.She turned wedding dresses into bold statements of female sexuality. If Madonna has been perpetually unapologetic (see: “Human Nature,” “Unapologetic Bitch”), it is partly because she ignited so much controversy at the beginning of her career by barely striking a match. Basically, she wore a wedding dress, while singing about the joys of sex, on a song called “Like a Virgin.” Also: She rolled around in the wedding dress while performing that song live on MTV. And the wedding dress wasn’t really your typical wedding dress, but a lace corset cinched by a belt that said “Boy Toy” on the buckle. And underneath she wore a white garter belt that was seen by a hefty percentage of America. —Jacob Bernstein

4.She announced her plans for world domination on national TV. Dick Clark asked a simple question: “What are your dreams, what’s left?” when Madonna appeared on “American Bandstand” in 1984. Her answer came quickly: “To rule the world.” —Caryn Ganz

The Life Picture Collection/Getty Images 5.She taught the world the words “Gaultier cone bra.” Elton John had his Tommy Nutter suits; Bowie had his Kansai Yamamoto jumpsuits. And now every worldwide arena tour has its full complement of custom designer finery, as if Beyoncé and Jay couldn’t possibly go On the Run without Gucci, Versace and Cavalli. But the apotheosis of pop star and pop couturier may well be Madonna and Gaultier, who made the now-famous cone bras and corsets for her Blond Ambition tour. Gaultier, whom the Paris fashion establishment continued to tag an “enfant terrible” long after his infancy, had been playing around with cone bras since the beginnings of his own label in the early ’80s. But Madonna gave them new life and worldwide infamy. They arrived in glimpses, peeking out through the slits of her Gaultier suits. And then the big reveal. She was boardroom bitch gone fertility goddess, a sex kitten with spine: Not for Madonna the flimsier side of lingerie. She remains his most famous client, and she, he has said, his best collaboration. “When you meet your idol, you can be disappointed,” he said when a retrospective of his work traveled the globe. “When I met her, I was not disappointed.” —Matthew Schneier

Michael McDonnell/Getty Images 6.She made the whole world dance. For those of us who think we know the real Madonna, she will always be a modern dancer. Not a singer who dances, but a dancer who sings. She dropped out of the University of Michigan School of Music, Theater & Dance, which she attended on scholarship, with a dream: to dance in New York. She hoped to join Alvin Ailey American Dance Theater and even earned a partial scholarship to its school, where she was told that she needed to study the Martha Graham technique. In an essay for Harper’s Bazaar, Madonna described it as being “physically brutal” with “no room for slouches.” It’s fitting that at the Graham school, you learn more than steps: The teachers, the students, the technique are all lessons in devotion, which is something that Madonna knows a thing or two about. Madonna always has been an exceptional dancer herself, blessed with a highly coordinated body and natural rhythm that allow her to get inside any groove. There’s something about her career, with its rigor and free-spirited abandon, that feels like a dance — singing may be her main artistic channel, but her performance is a passionate and ever-growing piece of choreography. Her gift to us has a ring of the eternal: She gave us back our bodies. —Gia Kourlas

7.She wasn’t Queen — or a Queen yet — but she ruled Live Aid anyway. In 1985, not long after Playboy and Penthouse published years-old nudes of her, Madonna arrived in Philadelphia for Live Aid overdressed for 95-degree heat (she looked like a refugee from Prince’s Revolution) and exuberantly performed semi-polished versions of “Holiday,” “Into the Groove” and “Love Makes the World Go ’Round.” When she told the ocean of people — some screaming “slut” — that she’d be keeping on her coat, the ocean booed. But Madonna turned spicy: “Nah. I ain’t taking [expletive] off today. They might hold it against me 10 years from now.” Queen is widely acknowledged to have been the most spectacular thing that happened that day (in the festival’s London block). But a lot of the world had never seen this brand-new pop star (and one of only a few women) sing live. In her way, she rocked us, too. —Wesley Morris

via Giphy 8.She lived every moment on camera in “Truth or Dare.” “Don’t hide back there, Warren,” she barked. “Get over here.” Who ever heard of an actor avoiding the spotlight? But Warren Beatty, bless him, was merely a movie star with nearly three decades of experience when he made his cameo in “Madonna: Truth or Dare.” Even movie stars need a moment when the camera stops. Not Madonna. She was a global superstar, at the height of her powers and in the thick of her Blond Ambition tour, starring in a documentary film she herself was executive producing. It takes a lot to relegate 1990 Warren Beatty to the background, but she did it. He was merely the boyfriend (they’d met filming “Dick Tracy”) and, in “Truth or Dare,” the scold. “Why would you say something if it’s off camera?” he mocks her later on, when she submits to a throat exam on film. “What point is there in existing?” “Truth or Dare” — which was that unlikely thing, a documentary box-office success — didn’t invent reality TV, with its on-camera exhibitionism, though it did prefigure our current moment, in which stardom means never having to (or getting to) tell the cameras to cut away. It didn’t invent the concert film, either. What it did was offer something of both: Madonna the show-woman and Madonna the holy mother, ministering to her brood of backup-dancer “children” — and the grit and sweat behind the two. She struts. She frets. She gives glimpses of the life that led her here. The world mocked her for it — there were parodies, and reviewers who sniped — but she peeled back the curtain, jumping from glossy, color concert footage to grainy, backstage black-and-white. She put up the money and then put her money where her mouth was: She gave the director, Alek Keshishian, a stripling of 26 at the time of its release, final cut. Who among today’s crop of pop starlets, with their death-grip control of their own mythologizing machines, would do the same? “One of the most surprising things was that she didn’t ask me to lose anything,” Mr. Keshishian said in an interview last year. “She only wanted to keep more.” —Matthew Schneier

Paramount, via Everett Collection 9.She brought yoga to the masses. Listen, did yoga really exist before Madonna? I know, I know. Some people will say it’s an ancient protocol. But if you were paying close attention, absolutely no one (O.K., no one I knew) was doing yoga until somewhere between 1998 and the early 2000s. Some attribute this to the internet, which helped spread new ideas. Or more open-mindedness to alternative healing. Those people are wrong. The only plausible reason for yoga’s mainstream infiltration is, of course, Madonna. Consider this: In her 1998 interview on “The Oprah Winfrey Show,” Madonna revealed that she no longer went to the gym — “I’m gym-free,” she said. “I’m liberated!” — and practiced only yoga. That same year, she appeared on “The Rosie O’Donnell Show” and demonstrated a sun salutation. Her album “Ray of Light” included a song called “Shanti/Ashtangi.” Then, in 2000, she starred in “The Next Best Thing” in which she just ... does yoga. Right there. For no great reason, in the middle of the movie. In a long, slow scene — I can remember watching, totally aghast — she moves through a series of poses. So that’s yoga, I thought. It was so singular an activity, so internal and quiet and complex and individual. It was like Madonna herself. Forget Bikram. Forget B.K.S. Iyengar. It was Madonna who gave us yoga. —Taffy Brodesser-Akner

via Giphy 10.She did more than almost anyone to reverse the idea that “disco sucks.” When Madonna released her first single, “Everybody,” in 1982 — a simple, synthy track produced by Mark Kamins, the D.J. from the nightclub Danceteria — it was not considered a smart plan for world domination. Disco music had been largely pioneered and championed by black and gay men, but the carefree sexual ethos of their late ’70s nightclub scene had given way to a reality slipping further to the right. (Remember “disco demolition night” and Anita Bryant?) So it wasn’t a big surprise that the divas of the early ’80s — Pat Benatar, Cyndi Lauper, Joan Jett — were queens of rock ’n’ roll. It was also part of why Madonna was written off as a flash in the pan. But Madonna never abandoned her devotion to the dance floor and her roots in disco. “Vogue” borrowed from the Salsoul Orchestra’s “Ooh, I Love It,” and Raw Silk’s “Do It to the Music.” Fifteen years later, “Hung Up” borrowed a riff from Abba. And live performances of “Music” and “Future Lovers” on her Confessions tour contained snippets of the Trammps’ “Disco Inferno” and Donna Summer’s “I Feel Love.” —Jacob Bernstein

11.She made us look at nudity from another perspective. When Madonna posed fully, frontally naked for “Sex,” her widely derided 1992 coffee table book, the shocking part wasn’t the nudity. It was the point of view: She subverted the viewer’s leering gaze by putting the focus squarely on her defiant character. The purity of Madonna’s vision, evident in every inch of the photo, defied any pornographic projection. There she stood, in a pose worthy of the most flagrant John Waters character — with cigarette dangling from a sneering mouth, as she stuck out a finger to hitchhike on an open highway. Every possible danger sign was invoked and inverted, staring down all forces that could contain her freedom. Other major female pop singers had done partial nudes before, including Joni Mitchell for the inside sleeve of her “For the Roses” album (shot from behind and at a significant distance) and Janis Joplin (with a tangle of necklaces covering much of her breasts, and her hands protecting what lies below). Yet both those images suggested degrees of innocence. Madonna’s closest precursor was Grace Slick, who mugged sardonically for several impromptu topless shots in the early ’70s. But those images established Slick’s power through her absurdist humor. In the “Sex” shot, Madonna used her body with as much animalistic force as her fellow Detroit rocker Iggy Pop. Her pose turned inside out the way sex is commonly used in pop culture nudes, expressing not a sexual lure but the power of personal confidence. —Jim Farber

via Giphy 12.She reclaimed crotch-grabbing for women. It’s a brief bit in the industrial-themed, “Metropolis”-inspired music video for “Express Yourself,” but it’s a standout. Madonna, in a double-breasted suit, is dancing on a platform at the top of a staircase. When she sings “Baby, ready or not,” she slowly slides her hand to her crotch and grabs it. It’s a move that Michael Jackson made famous, but when she does it, it’s an act of empowerment, a celebration of women taking ownership of their sexuality. —Mekado Murphy

13.She reflected on, and interrogated, her influences and work in a landmark interview. Madonna gave a gigantic interview, in 1999, still at the apex of her creative genius, to the critic Vince Aletti — not for Rolling Stone or The New Yorker, but Aperture — about her tastes in photography, painting, gender, sex and a bunch of other things. It’s a gold-standard conversation. She’s hypnotically, casually frank here. Mr. Aletti respects her, which frees her brain to rove. He asks, “When I think of you and males, I think of all the guys in your videos, most of whom have been like thugs.” She replies: “Hunky boys? Yes, I am attracted to a thug. I like that quality, but I like the other side of it, too. Because all guys who go around behaving in macho ways are really scared little girls. So you have to look beneath the surface. There’s a difference between my ideal man and a man that I’m sexually attracted to. Believe me. Therein lies the rub.” Nowadays, nobody talks about their work and influences — about themselves — as vastly as she does. And that’s one page. There are 16 more just like it. —Wesley Morris

Frank Micelotta/Getty Images 14.She became pop’s headmistress of reinvention. Madonna’s ability to transform herself for albums, videos, photo shoots, tours and movies made her a master of perpetual transformation. As she rolled from one aesthetic to the next, switching up her hair, her choreography, her accent, it became shorthand to describe what was happening in one word: reinvention. For a while, Madonna rolled her eyes at the description. It seemed crass and calculating in a way she didn’t like. She’d long admired Cindy Sherman, an artist whose chameleon-like quality seemed to her not to be a signal of cynicism, but curiosity. But after fans complained about the lack of hits performed on her 2001 Drowned World tour — Madonna does not like to revisit the past — she decided to pay tribute to her many artistic periods by calling its follow-up tour Re-Invention. Later, she credited her understanding of the concept to the fashion photographer Steven Meisel and the shoot they did together for the album cover of “Like a Virgin.” “He treated each photo shoot like it was a small film and insisted that we create a character each time,” she told Vogue, “but then would make fun of the archetypes we created.” —Jacob Bernstein

15.She sparked major conversations about race — and cultural appropriation. “Art should be controversial, and that’s all there is to it,” Madonna declared to The Times in 1989, discussing her “Like a Prayer” video. The song initially had its premiere as part of a Pepsi TV ad, and Madonna released the full music video the next day. Sharing only an African-American gospel group in common with commercial, the five-minute video showed Madonna running into a church after she witnessed a group of white men assaulting a white woman, having a spiritual and sexual orgasm in the church’s pulpit, and coming forward to testify on behalf of the African-American man who was wrongfully arrested and incarcerated for the crime. The video ignited protests over its religious imagery, especially her stigmata scene, the statue of a black saint and the burning crosses. The director Mary Lambert later reflected, “I knew that we were pushing some big buttons, but I sort of underestimated the influence and bigotry of fundamentalist religion and racism in this country and the world.” Under the threat of boycott, Pepsi capitulated by dropping the $5 million ad campaign and its sponsorship of her world concert tour. Madonna was applauded by some for breaking racial and religious taboos, but the video also initiated a critique that not only haunts much of her later work and life, but also heralded public debates about the abuse and misuse of African-American cultural identities by white artists. Remarking on the video in 1992, the feminist critic Bell Hooks wrote, “Fascinated yet envious of black style, Madonna appropriates black culture in ways that mock and undermine, making her presentation one that upstages.” Looking back, “Like a Prayer” did not simply announce Madonna as our cultural provocateur, but as an iconic cultural appropriator as well. —Salamishah Tillet

Jeff Kravitz/FilmMagic, via Getty Images 16.She tangled with the sharpest man in late-night TV. It takes a lot to ruffle David Letterman, but in a 1994 appearance, Madonna did her best to offend the host of the “Late Show” — swearing profusely (“You realize this is being broadcast, don’t you,” he quipped), calling him a liar and ultimately refusing to leave the stage. Several months later, the pair made a public appearance at the MTV Video Music Awards to suggest all was well, and Madonna returned to Letterman’s show several times after that. In 2005, when she was recovering from serious injuries from being thrown from a horse, Mr. Letterman had two waiting outside the theater, offering her the chance “to get back on the horse.” They trotted down 53rd Street together, trading barbs. —Caryn Ganz

Kevin Winter/Getty Images 17.She dared to be a sexual woman … who still continued to grow older. Women don’t get to be sexual in old age. Or so goes the adage — and the “Inside Amy Schumer” sketch where Julia Louis-Dreyfus toasts her last day before she isn’t desirable anymore. So while Mick Jagger and Rod Stewart happily thrust their way into their 70s — and Lenny Kravitz can make dad-jokes about his penis falling out of his pants — female artists field questions about “aging gracefully” and “acting their age.” Camille Paglia advised that Madonna give up the “pointless provocations” and “trashy outfit[s]” of her “prolonged midlife crisis.” When the singer twerked on “Carpool Karaoke,” the television host Piers Morgan scolded: “You can’t be 58 and dancing around like that.” He then pretended to vomit into a bucket. If you’re Madonna, though, you will continue to provoke by simply daring to … be an aging woman who exhibits desire (and a need to be desired). Which is almost comical, since Madonna also manages to defy physical signs of aging. But perhaps that, too, is part of the calculation: the ability to conform, just enough, to what the industry demands while simultaneously crusading to end the ageist — and sexist — structures that require it. —Jessica Bennett

via Giphy 18.She made ’90s electronica mainstream with “Ray of Light.” Every generation gets the iteration of electronic music it deserves. In the ’90s, a new synth sound was packaged under the “electronica” banner, pioneered by acts like the Prodigy, the Chemical Brothers and Moby. Reprising her role as the ultimate cultural conduit, Madonna sifted the best from the e-world, put it into a more melodic context and so, brought the sound to a whole new audience. —Jim Farber

Gareth Cattermole/Getty Images 19.She never stopped flexing. Of course Madonna took heat for being too muscular. (Was there anything that wasn’t too much for somebody?) Her fitness, flexibility and strength have always been tied to the kind of cultural power she wields: She defined a standard for how strong a pop star could look, and few have risen to the challenge. Yes, we know she does yoga and Pilates. She is also a partner in a chain of health clubs called Hard Candy Fitness and could probably crush your head in one of her chiseled biceps. —Caryn Ganz

20.She was one of the first major pop stars to leave a traditional record deal behind. Madonna released her first album in 1983 under a division of Warner Bros. Records and maintained a 24-year career there, selling over 200 million albums worldwide. But in 2007, at 49, she did something bold and new: She left the label and signed with a concert promoter. Live Nation offered a 10-year partnership worth more than $100 million that included studio albums and merchandising, but mostly relied on the artist as a touring powerhouse. It was a sign of how musicians would move forward in a digital age when albums were becoming more and more difficult to sell. —Mekado Murphy

Frank Micelotta/Getty Images 21.She was early to dabble in lesbian chic. Tired: Madonna and Britney at the 2003 Video Music Awards. Wired: Madonna and Ingrid Casares!!! —Matthew Schneier

via Giphy 22.She poked fun at herself in a New Yawk accent. In an unforgettable 1992 “Coffee Talk” sketch on “Saturday Night Live,” Madonna held her own as a Yiddish-speaking, Barbra Streisand-loving friend of Mike Myers’s Linda Richman. (Inside joke: Her character, Liz Rosenberg, shared a name with her loyal, longtime publicist.) “Eccch, I don’t like that Madonna,” she groused to one caller. “She’s a tramp — every week with a different boyfriend.” Only the arrival of the real Barbra at the sketch’s end could top it. —Scott Heller

via Giphy 23.She gave MTV’s Video Music Awards their defining tone. When Madonna rolled around on the floor of the Radio City Music Hall stage in a white wedding dress moaning “Like a Virgin” on the very first V.M.A.s in 1984, jaws dropped around the world, instantly putting the event on the map. Each year since, everyone performing there has strained to make that starry a splash. (Madonna did, herself, in 1990, bringing “Dangerous Liaisons” chic to MTV and striking very memorable poses in a bewigged version of “Vogue.”) —Jim Farber

Amos Gumulira/Agence France-Presse — Getty Images 24.She did more than pay lip service to vulnerable children in Africa. Celebrities donate money at benefits and serve as the face of charities. But Madonna’s dedication to bettering the lives of at-risk children in Malawi is an indelible part of her life. She adopted David Banda from an orphanage there after meeting him in 2006, returned in 2009 for Mercy James, and again in 2017 for the twins Estere and Stella. She co-founded Raising Malawi, a nonprofit focused on helping the county’s orphans by supporting education and health initiatives, and made a documentary, “I Am Because We Are,” to raise awareness about children’s needs there. Her efforts haven’t been drama-free — there was an abandoned school project, and public disagreements with David’s father — but she’s remained dogged. And for her 60th birthday, she asked fans to do one thing: support a Raising Malawi fund-raiser. —Caryn Ganz

Buena Vista Pictures, via Photofest 25.She tried to make Broadway — yes, Broadway — cool. By the time the movie musical “Evita” came around in 1996, Madonna had already done Broadway — she originated the role of Karen in David Mamet’s “Speed-the-Plow,” for God’s sake — but there’s Broadway and then there’s Broadway. Bless her, who has ever pushed harder and spent more hard-won cachet trying to make musical theater cool? She took the singing lessons. She did the media rounds (she appeared on the cover of Vogue in October 1996 in full “Evita” drag). Then she returned to pop, and there she stayed. But she does have one other indelible musical theater performance, even if it’s often overlooked: “Sooner or Later” from the movie “Dick Tracy,” written by none other than Stephen Sondheim. If she didn’t win him the Oscar for it, she didn’t hurt his chances, either. And to see her, selling it onstage at the Academy Awards in full bombshell splendor — shucking off a glove and tossing aside a stole, vamping at the end for longer than seems humanly possible — is to see a way the Great White Way wouldn’t, and didn’t, know what to do with. There’s Broadway and then there’s Madonna. —Matthew Schneier

Stephen Lovekin/Getty Images 26.She made gap teeth — yes, gap teeth — cool. That smile. The looks around it changed — pop, goth, cowgirl, plastic-fantastic — but there’s always been that Madonna mouth. Maybe I speak as one who took a lax approach to the stern recommendations of Big Orthodontics, but who worked a slight space between the teeth better than Madonna? She’s the best P.R. for the sensual power of gap-teeth since the Wife of Bath back in the 14th century. —Matthew Schneier

Cory Schwartz/Getty Images 27.She wickedly played with the idea of another pop star plagiarizing her song. In 2011, there was a brief kerfuffle over the chorus of Lady Gaga’s “Born This Way” sounding like the famous refrain of Madonna’s “Express Yourself.” Madonna stayed mum at first but made a big statement on her 2012 MDNA tour, performing a mash-up of the two songs while dressed like a cheerleader and twirling a baton. —Caryn Ganz

Gerard Julien/Agence France-Presse — Getty Images 28.She made innerwear outerwear. In “Desperately Seeking Susan,” Madonna reinvented the bra and panties (as a bikini), and stalked a Port Authority Bus Terminal bathroom in her bra. She played an exotic dancer in her “Open Your Heart” video, wearing a black corset with sparkly gold pasties on top. And who could forget the missile-like cone bra of Blond Ambition, which helped establish that in Madonna’s work, sex was less often a thing to be celebrated in and of itself than it was an extra avenue for power. —Jacob Bernstein

29.She had one of the best diva moments ever — over a hot room. There is a subgenre of videos online dedicated to compiling Madonna’s most glorious diva moments. In the one most of my friends have memorized, the singer complains that she can’t do an interview in a stuffy room. “I’m suffocating,” she says. “Just give me a fan.” The clip ends with her taking the journalist’s sheet of questions … and fanning herself. —Caryn Ganz

30.She has writing credits on nearly every one of her hits. Take a look at the liner notes on every Madonna album since her third LP, and you’ll notice her name popping up. A lot. That’s because she has a writing or production credit on all the songs on her records starting with the 1986 release “True Blue.” You can call it a function of her need for creative control — she’s had input on every track; you can call it a savvy business maneuver — but it guarantees her royalties on each hit. You simply can’t say she hasn’t put her fingerprints on nearly her entire body of recorded work. —Caryn Ganz

via Giphy 31.She helped transform Marilyn Monroe from a symbol of victimhood to a spokeswoman for unapologetic sexuality. In 1985, Madonna channeled Marilyn Monroe circa “Gentlemen Prefer Blondes” in her “Material Girl” video. Monroe at that point was in many ways a tragic figure, a woman whose death from barbiturates cemented her status as the sad, sexy girl. That wasn’t Madonna’s interest. With men dancing around her, begging for affection, Madonna made her tribute a statement of sexual assertiveness and power. —Jacob Bernstein

32.She put the “M” in MTV. Madonna’s clips have given the network more media headlines than any other artist. And no other performer has used music video to serially elaborate their personas with a finer mix of creativity, beauty and edge. While Michael Jackson went the more-is-more route, spending additional money, extending the lengths and blowing stuff up, Madonna deepened the richness of the images through gorgeous pieces like “Open Your Heart”; widened the breadth of her concepts (“Frozen”); and emboldened her content (more than one of her videos has been deemed unsuitable for broadcast). She worked with Herb Ritts (“Cherish”), David Fincher (“Express Yourself”), Mark Romanek (“Rain”) and Jonas Akerlund (“Music”), and many others lucky enough to have her name among their credits. —Jim Farber

33.She was a pioneering ally to L.G.B.T. people everywhere. These days, there’s barely a female pop star alive who doesn’t loudly broadcast her unwavering, if sometimes exploitative, support for L.G.B.T. rights. But in 1991, when Madonna gave her no-holds-barred interview to The Advocate, then the largest voice of gay communities, she showed more understanding of queer issues and identity that any pop star before her, and most who came after. In the two-part sit-down, Madonna revealed the roots of her gay identification via her early mentors. It wasn’t just empathy she was expressing but identification, forged by her defining, early experiences in gay clubs. Important, too, was the interview’s timing, published during the peak era of H.I.V. infections and deaths. It was a time when many celebrities spoke about gay people only in the most nervously chaste ways. Madonna, who had earlier released a public-service announcement about safe sex, went further. She endorsed the uncompromising activist group Act-Up, and spoke in sex-positive terms with a comfort and care that was decades ahead of its time. —Jim Farber

34.She had a secret life as a downtown rock star. Madonna didn’t learn to play guitar until her “Music” era, but rock bands were fascinated by her much earlier than that. Mike Watt of the Minutemen had a spinoff devoted to interpreting her songs called the Madonnabees, and the Sonic Youth side project called Ciccone Youth released “The Whitey Album” in 1989. In a recent Guardian piece, Thurston Moore recalled how Madonna was thoroughly plugged in to the downtown scene when the two were neighbors in New York. “Madonna was actually in a couple of no-wave bands that nobody ever talks about,” he said. Time to dig those up. —Caryn Ganz

Sandy Hill/Associated Press 35.She set the tone, and the bar, for modern pop megatours. Donna Summer and Michael Jackson had both fused popular music and theater in service of live performance, but the scale of Madonna’s 1990 Blond Ambition tour was bigger, bolder and more imaginative. There were seven dancers and two backup singers, not to mention her band. There were a variety of sets (including a “Metropolis”-inspired one) and costumes designed by Jean Paul Gaultier. “Truth or Dare” helped secure Madonna’s place as the principal architect of the contemporary pop spectacle, a person Miley Cyrus, Katy Perry and Taylor Swift have turned to for inspiration. —Jacob Bernstein

36.She made talent scouting a central part of her art. Madonna may have referenced spending some time “as a narcissist” on her 2015 song “Rebel Heart,” but curiosity about others has always been the fuel propelling her forward. She has had a gift for finding the right people to complement her own skills and open up her world: the fashion designers and stylists, the video directors, the D.J.s and producers, the photographers, the cool hunters, the romantic partners. Describing her own ambitions in a 1989 Rolling Stone interview, she summarized it well: “I would rather own an art gallery than a movie studio,” she said. “Or a museum. I would rather be Peggy Guggenheim than Harry Cohn.” —Jacob Bernstein

via Giphy 37.She brought vogue from the ballroom to the living room. Madonna wasn’t the first pop star to plunder the grace and hauteur of vogue balls. The serial exploiter Malcolm McLaren beat her to it with his “Deep in Vogue” single in 1989. Yet, less than one year later, Madonna drew so creatively from that rich demimonde that she wound up providing the ideal conduit between it and Every Mall U.S.A. In her “Vogue” clip, Madonna featured experienced vogue dancer/choreographers from the Harlem “House Ball” community, including Jose Xtravaganza and Luis Camacho, who had introduced her to the style at the New York club Sound Factory. She then hired them as virtual co-stars on her Blond Ambition tour. The resulting smash appeared a year before Jennie Livingston’s documentary “Paris Is Burning,” which filled in the background of the balls. However much cultural appropriation went into “Vogue,” Madonna’s take on it had to strike anyone, from inside of that world or out, as fierce. —Jim Farber

Getty Images 38.She found more than one way to aggravate the Vatican. Madonna’s fascination with Catholicism has been well-documented (on this list, alone!). But Catholicism has kept a close watch on Madonna, too. The Vatican condemned her “Like a Prayer” video (obviously), and Pope John Paul II urged a boycott. In 2004, the church wasn’t crazy about her embrace of kabbalah, and when her Confessions tour made its way to Rome — with Madonna singing “Live to Tell” tethered to a disco-ball cross — she was accused of blasphemy. “It is nothing short of a scandal and an attempt to generate publicity,” said Cardinal Ersilio Tonino, who seemed to be catching on. —Caryn Ganz

Jeff Kravitz/FilmMagic, via Getty Images 39.She dressed up like a clown and performed a show. Twice. If you thought Madonna was out of ways to shock people by 2016, her Tears of a Clown show would have been an eye-opener. For two performances only (so far), in Australia and Miami, Madonna dressed up like a (very high-fashion) clown — makeup, tricycle, the whole shebang — and performed a night of bizarre comedy, covers and stripped-down arrangements of her songs. In Australia, she played a song by Elliott Smith; in Miami, Britney Spears. (Yes, I was there, and there were tears.) —Caryn Ganz

40.She made the hiring of stylists standard operating procedure. Sometime in the early ’80s, Madonna was introduced to the fashion maven Maripol at a hip-hop show at the Roxy. Stars had worked with fashion designers. “But they were couturiers,” Maripol, who was previously the art director of the downtown store Fiorucci, told me. “The concept of a stylist didn’t really exist.” Maripol began putting rubber bracelets on Madonna’s wrists and helped construct the look for the “Like a Virgin” album cover. Part of their bond came from their Catholic upbringings: Both were propelled by a love for its iconography and an aversion to its conservative teachings. Madonna moved on, but her decision to work with stylists — among them Paul Cavaco and Lori Goldstein — stuck. Since 1998, her principal stylist has been Arianne Phillips, who helped put together the gothic geisha and Raphaelite yogi looks of the late ’90s “Ray of Light” era and then collaborated with her on every tour after that. Meanwhile, a new generation of singers and actresses borrowed from Madonna’s playbook and got stylists of their own. They’re now ranked annually by The Hollywood Reporter. Ms. Phillips doesn’t appear on it. She’s become an in demand costume designer with two Academy Award nominations under her belt. —Jacob Bernstein

via Giphy 41.She was a part of the most memorable V.M.A.s red carpet interview. Kurt Loder was asking generic questions about Madonna’s new album at the 1995 MTV awards when an impostor made her presence known by lobbing a projectile into the frame. Should we let Courtney Love come up, Mr. Loder wondered? “No, don’t, please,” Madonna said. But it was too late. What followed was four minutes of gloriously awkward back and forth about Alanis Morissette, Dennis Miller, their designer shoes and Kurt Cobain’s take on “Truth or Dare.” —Caryn Ganz

42.She was the first great identity artist. Because Madonna treated the genders and other people’s identities as fashion, she made those identities seem fashionable. The line is fine between lifting a rock on a scene and a five-finger cultural discount. But there was always something amazingly generous in, say, putting “Vogue” on the “Dick Tracy” soundtrack and conjoining these nonwhite dancers with the white movie stars in the song’s bridge: It’s all fluid, so “go with the flow.” She didn’t care about history or socioeconomics — not in the music — only the justice in optics. I’d seen her rolling around in a flamenco dress in “La Isla Bonita” before I’d ever experienced Menudo. I saw the dancers in “Truth or Dare” and the video for “Vogue” before I actually knew I was gay. (Maybe I knew I was gay because of “Truth or Dare.”) Today, we’d mock her for the cornrows in “Human Nature.” We’d protest her playing Eva Perón. But identity was a game that nobody played better — or with more affection, perversity and provocation — than the white lady in the flamenco dress. —Wesley Morris

43.She spoke frankly about having had abortions. In 1986, Madonna removed the fingerless gloves and plastic bracelets, coifed her hair and started channeling Doris Day for “Papa Don’t Preach,” a song about a pregnant teen who was keeping her baby. But Madonna was among the first major stars to unapologetically admit to having had abortions. She did it matter of factly and without explanation in a 1994 interview with The Face when the backlash from her “Sex” book put her in the penalty box. At that time, only a handful of big stars (see: Sinead O’Connor) talked openly about their personal experiences with abortion. And it was several more years before celebrities like Whoopi Goldberg, Chelsea Handler and Stevie Nicks did the same. —Jacob Bernstein

44.She sullied the reputation of entire species of flower with a single off-handed comment. “I absolutely loathe hydrangeas,” she sighed into a hot mic, and hydrangeas were never the same. —Caryn Ganz

via Giphy 45.She sullied the reputation of Kevin Costner with a single off-handed comment. It happened when the actor’s backstage visit was captured in “Truth or Dare.” What did he think of her Blond Ambition show? “I thought it was neat,” he said. She gagged behind his back. Mr. Costner later said he was “hurt,” but he did appreciate an apology she issued from the stage a decade later. —Caryn Ganz

via Giphy 46.She promoted erotically charged images of same-sex couples. Today, same-sex couples are as much a motif in music videos as choreography. Not so much in 1990, when Madonna didn’t just represent the eros of the community, she put herself right in the action. In the “Justify My Love” video, she presented her first lesbian kiss, 13 years before she locked lips with Britney and Christina. In 1991, during her “Truth or Dare” documentary, she had two of her male dancers smooch in what several generations of gay men reported was the first time they had ever seen their own desires represented so friskily onscreen. —Jim Farber

47.She became briefly British. Madonna took what she needed from a move to London — a hunky husband and a quasi-British accent. But when she got sick of the local tabloids calling her Madge, she returned to America. The accent came with her for a while. —Scott Heller

via Giphy 48.She turned an MTV ban into cash. MTV declined to play her steamy “Justify My Love” video, so Madonna did the next best thing: released it on home video as a “video single.” The cost: $9.98. —Caryn Ganz

49.She spun an Abba sample into her own disco brilliance. Abba is famously averse to granting other artists permission to use its samples, but the Swedish group allowed Madonna to use a hook from “Gimme! Gimme! Gimme! (A Man After Midnight)” on her 2005 track “Hung Up.” The song (and its outstanding video, which did a lot for pink leotards) announced the arrival of “Confessions on a Dance Floor,” one of her greatest albums, and gave Madonna her 36th Top 10 hit. —Caryn Ganz

via Giphy 50.She made her backing singers and dancers household names. They leapt to life as personalities in “Truth or Dare,” and some of them continued to appear alongside Madonna on tour and in her videos — the backup singers Donna De Lory and Niki Harris, and the troupe of dancers: Luis Camacho, Oliver Crumes, Salim Gauwloos (known as Slam), Jose Xtravaganza, Kevin Stea, Gabriel Trupin and Carlton Wilborn. The crew was such an integral part of Madonna’s Blond Ambition tour, fans learned their names and stalked them for autographs. Last year, the dancers were even the focus of their own documentary, the revealing “Strike a Pose.” —Caryn Ganz

51.She salvaged a potentially disastrous turn at the Tonys. Madonna demonstrated how to futz your way through an awards-show presentation with aplomb after stumbling through the regional-theater prize at the 1988 Tony Awards. (Side note: Who decided Madonna should present that?) “I’m being punished for not coming to rehearsal today,” she coyly confessed before inviting the winners onstage. —Scott Heller

52.She provided a key role model for sex-positivity for several generations of women. From Mae West to Susie Bright, pop culture brims with entertaining advocates for female sexual assertion. But few have made it their mission to push it as consistently, loudly and resonantly as Madonna. She has been on message from Day 1, not only talking about the power of sex, but also displaying as many examples of it as blatantly as she possibly could. —Jim Farber

Ron Galella/WireImage, via Getty Images 53.She made crucifixes cool. In her 1991 interview with The Advocate, Madonna called the religion of her birth “mean.” Yet, by her melding one of Catholicism’s central images with her personal style, she repurposed the crucifix as a symbol of acceptance and play. —Jim Farber

54.She had a rare run of five great albums in a row (and then some). Madonna had a five-album, great-album streak that lasted from 1989 (“Like a Prayer”) to 2000 (“Music”) and included “Erotica,” “Bedtime Stories” and “Ray of Light.” She had a letdown with “American Life” in 2003, when her ambition — and her rapping — seemed to get the best of her. But she rebounded in 2005 with “Confessions on a Dance Floor,” then “Hard Candy” (when she was 50), “MDNA” and even “Rebel Heart,” all of which are better than you think, and part of the reason for their being underrated is the exact occasion for this birthday party. We don’t know what to do with female artists — especially pop stars — still great in middle age who refuse to phone it in, in a stadium or the studio. She knows the culture thinks she should be taken out back and shot, which is maybe why the gun has become almost as prominent an image in her work as the crucifix. She’d rather shoot first. —Wesley Morris

55.She helped bring you “You Oughta Know.” Maverick, the record label Madonna helped found in 1992, had run its course by the early 2000s. But one of its 1995 releases is still making waves today: Alanis Morissette’s “Jagged Little Pill.” Ms. Morissette’s music found its way to Guy Oseary (Madonna’s longtime manager and business partner) who played Madonna the tape. “We were just blown away immediately by a) her songwriting skills and b) her voice,” Madonna has said. “We knew right away she was going to do well.” —Caryn Ganz

Stuart Wilson/Getty Images 56.She helped (briefly) make kabbalah the “it” religious movement. By sporting a red ribbon on her wrist, she (fleetingly) put the faith in vogue. —Jim Farber

57.She scored more Top 10 hits than any other artist. Madonna’s 38 Top 10 hits on the Billboard 100 singles chart have put her in rarefied air (the Beatles are behind her, with 34). The list includes the usual suspects and some fan favorites: “Secret,” “Deeper and Deeper” and “Don’t Tell Me,” as well as “Give Me All Your Luvin,” the song that caused a stir at the 2012 Super Bowl halftime show when M.I.A. extended her middle finger during her verse. And Madonna remains one of the highest-grossing live acts, too. Her Sticky & Sweet tour supporting “Hard Candy” brought in $408 million in revenue, and her overall ticket sales passed $1 billion eight years ago. —Caryn Ganz

58.She turned her fans into Mini-Mes. There were more than a few young folks parading around in one sequined glove in the ’80s. But a trip to a Madonna concert yielded a cornucopia of women (and men) dressed like the Queen of Pop — largely because she provided her devotees with so many options. Fans in veils? Sure! Fans in armloads of plastic bracelets? Yes, they still show up! Today it’s common to see audiences draped in their favorite star’s garb from album covers, videos and photo shoots. But as she so often did, Madonna set the standard. —Caryn Ganz

59.She proved herself to be the queen of media damage control. Madonna gave the world a lesson in how to defuse a potential scandal in 1985 after crummy nude images, shot early in her career, were sold by their photographer for exploitative layouts in Playboy and Penthouse. In a move that suggested she could have a nice second career as media consultant, Madonna announced, “I’m not ashamed” on the eve of the photos’ appearance, giving her control of the story while also reclaiming full ownership of her body. —Jim Farber