It’s hard to make a comeback in Hollywood — unless you’re swine.

In which case, the industry may embrace you. Just ask Miss Piggy. Nearly 40 years after she made her showbiz debut, the diva has never been more in demand. She recently appeared on the final episode of Jimmy Fallon’s “Late Night,” singing a melancholy song with her Muppets pals; dished about her love life on “The Bachelor”; and sipped Lipton tea in an Oscars ad. The weekend of the ceremony, she popped up on an E! fashion segment, dressed in a frock designed by Vivienne Westwood, and attended a pre-Academy Awards QVC party, where she entered via cabana lifted by four shirtless hunks. Then the situation got a little sticky. According to tabloid reports, Piggy shoved Joan Rivers into a cake backstage, and Rivers emerged at the party with frosting in her hair.

Like any shrewd publicity-seeking celeb, Miss Piggy is simply promoting her upcoming movie. In “Muppets Most Wanted,” which opens March 21, Piggy goes head to head with another diva — Celine Dion, who makes her bigscreen debut. In a showstopping number at the climax of the film, the pair belt out “Something So Right,” an emotional ballad. “I’ve had the opportunity to sing with some of the greatest voices of all time,” Dion says, “but none of them could compare to Miss Piggy.” When asked to describe her collaborator’s voice, she assesses: “Very sensual.”

The world is back in love with the Muppets, 59 years after their creator, Jim Henson, debuted his puppets, and a decade after the Jim Henson Co. sold them to Disney for an undisclosed sum. The partnership has been beneficial for both parties. “We feel the Muppets at Disney are a big success,” says the late puppeteer’s daughter Lisa Henson, who runs the Henson Co. in Hollywood with her brother, Brian.

At the height of their fame, the Muppets were the Pixar characters of their time, headlining entertainment targeted to both children and adults. Their first feature, 1979’s origins story “The Muppets Movie,” was a smash hit, and by 1984’s “Muppets Take Manhattan,” the biz-focused Kermit the Frog and his gang huddled on a Central Park bench, browsing the want ads in Variety.

But as the new millennium began, the Muppets had lost their mojo, unable to compete with a new crush of children’s entertainment programming, such as Pixar’s “Toy Story” sequels, “SpongeBob SquarePants” and Hannah Montana. Then in 2011, after seven years as the Muppets’ new owner, Disney rebooted the characters in a live-action film starring Jason Segel and Amy Adams that grossed $165 million worldwide. The movie claimed another franchise milestone, winning the Oscar for original song (“Man or Muppet”) — and prompted the studio to greenlight a sequel.

Suddenly, the Muppets were hot again.

The stakes for “Muppets Most Wanted,” the eighth theatrical movie starring Kermit and Co., are especially high. The $50 million adventure, which co-stars Ricky Gervais, Tina Fey and Ty Burrell, will prove if the Muppets are really here to stay. “The 2011 movie did exactly what it needed to do,” says Brian Jay Jones, author of “Jim Henson: The Biography.” “I think it’s the next one that really counts. (Puppeteer) Frank Oz said that what really makes the Muppets work is affectionate anarchy.”

So far, signs are encouraging, according to executives at Disney. Sean Bailey, the studio’s production president, counts his 9-year-old son as part of the new generation of fans who have recently discovered the Muppets. “I think these are pretty special characters who deserve to be around,” Bailey says. His production executive, Kristin Burr, is among those who see the property benefiting from the generational shift.

“Kids didn’t know who the Muppets were, because they weren’t around before the last film,” notes Burr, pointing out that now the characters are so ubiquitous it’s hard to turn on the TV without catching a glimpse of their mayhem. A potential reason for that: “The publicity team says it’s easy to book the Muppets on talkshows, much easier than movie stars,” says producer Todd Lieberman.

Kermit’s upcoming press tour will include a stop at “The Tonight Show,” on the same stage where he made his national debut in 1956 opposite Steve Allen. “I’m going to interview him like I’d interview any talking frog,” Fallon vows. “I’m going to ask him the hard-hitting questions.”

They may do a skit, too, for the St. Patrick’s Day appearance. “There could be something green happening — wink, wink,” Fallon says. The new “Tonight Show” host is one of the Muppets’ biggest admirers, having booked them on “Late Night” many times over the years. Their performance of “The Weight,” which played him off his former talkshow, has earned 2.2 million views on YouTube.

“Muppets Most Wanted” is a crime caper that picks up where the last film in the franchise left off, with the reinvigorated Muppets ready for a second act. Gervais, playing their shady manager, convinces them to embark on a world tour. His sidekick is a new Muppet, Constantine, a criminal frog from Russia who looks awfully similar to Kermit, with the exception of a mole on his snout. Gervais felt an immediate bond with his co-star. One day, while the cameras were setting up, he recalls asking Constantine what he ate for lunch. “Chicken,” came the reply (with the help of puppeteer Matt Vogel). “Do frogs eat chicken?” Gervais wanted to know. “Yeah,” Constantine shot back. “Do frogs eat french fries?” Gervais pressed on.

“And I look over,” Gervais recalls, “and the extras were looking at us, thinking, ‘No one is filming, and he’s talking to a frog now, without laughing or goofing around; he’s just having a chat about what they had for lunch.’ ” Gervais, on a phone interview from London, is cracking himself up. “I heard those stories about how Drew Barrymore thought E.T. was real when she was little,” he says. “They thought that maybe I’m like that, but no one has the heart to tell me the frog isn’t real.”

Kermit was equally alive for Fey, who plays a prison guard in a Russian gulag who develops feelings for the amphibian. In an off-camera scene, she tried to stand in his eye-line when reading dialogue, only to be told she could sit down, since the puppet couldn’t see her.

In more ways than one, the Muppets have been entertainment pioneers. When Jim Henson launched “The Muppet Show,” which ran for 120 episodes from 1976 to 1981, he booked celebrity guests like Elton John, Steve Martin, Liza Minnelli and a 15-year-old Brooke Shields. “I actually made some of the characters for the crowd scene in the workshop,” recalls Shields, who appears in a classic “Alice in Wonderland”-themed episode. “My mom had always been a fan of the Muppets.” So was a young James Bobin, the English director of the current film and its predecessor, who co-created the quirky HBO series “Flight of the Conchords.” “I’m pretty sure,” he says, “the Muppets are good ground for what I find funny now.”

As the Muppets became a global phenomenon, they shaped a generation of future comedians. “I call the Muppets the gateway drug to comedy,” says screenwriter and director Nicholas Stoller (“Forgetting Sarah Marshall”), who co-wrote the two most recent Muppets films. “It’s the first thing you see as a kid.” Gervais concurs, saying he might have gotten the idea of “Extras” from the franchise: “A bunch of Muppets bringing down self-important people,” he explains. Fey says that in the middle of making “30 Rock,” she read an essay arguing that her comedy series borrowed everything from Henson’s world. “They broke it down character by character, and I was like, oh my God, it’s kind of true,” she says. “Jenna is like Miss Piggy, Jack Donaghy is Sam Eagle, and I guess maybe I’m Kermit. I didn’t mean to rip it off.”

Gervais remembers tuning in to “The Muppet Show” every Sunday at 5 p.m. from the U.K., where the series was filmed, after no U.S. network would gamble on a puppet-driven series. Burrell says he watched from his childhood home in Applegate, Ore., while Fey was glued to her set in Pennsylvania. “It was a great show, because your parents would watch it too,” Fey says. “I remember being excited waiting for ‘Muppets Take Manhattan,’ because there were these Burger King tie-in glasses, and that was the highlight of my summer.”

Gervais describes his role in the movie, without a hint of irony, as a dream come true. “I’ve always been slightly jealous of Michael Caine, not because of his other films, but because he was in ‘The Muppets Christmas Carol,’ ” he says, referencing the 1992 adaptation of the Charles Dickens story starring Kermit, Miss Piggy, Gonzo and the other characters created by Henson. “I’ve watched him for 20 years, going, ‘Damn that man — he was the lead in a Muppets movie!’”

Burrell, who plays a bumbling detective in the upcoming film, was also “over the moon,” to be a Muppets co-star. “I was like, really me? And they don’t know this, but it could have been the easiest negotiations of their entire lives. They could have offered me like $1.50 and some paper-towel coupons, and I would have done the Muppets.”

Bobin offers some trade secrets. The Muppets stage is often raised four feet off the ground, so the puppeteers can stand and watch themselves performing on monitors, a system Henson perfected when he first started appearing with puppets on a local NBC TV station in the ’50s. “There aren’t very many Kermits, because he’s a very malleable well-built puppet,” Bobin says. “Piggy’s head is made of a weird foam material, kind of like a cast, so when she talks a lot, her mouth starts to crack in the inside. In the entire movie, we must have gone through 30 Piggy heads.” Much like her character, it seems, Miss Piggy needs a lot of attention.

In 1970, Kermit made television history when he appeared on “Sesame Street,” where he sometimes popped up as a friend of Big Bird, and strummed a ditty called “Bein’ Green.” But the lyrics to what would become his signature song, “It’s not easy bein’ green,” might as well have been redubbed later in his career to, “It’s not easy making green.” After Jim Henson died suddenly at age 53 from a streptococcal infection, the Muppets struggled to stay afloat. Henson had been in prolonged negotiations to sell his catalog to Disney, where he hoped it would be in safe hands. “My father did want the Muppets to be at Disney,” Lisa Henson says. “He was in love with Disney World, and everything was so meticulous about how Disney ran the parks. But unfortunately, he died just as the transaction was closing.”

With the Henson family continuing ownership, the Muppets kept plugging away. Brian directed two ’90s chestnuts, “The Muppets Christmas Carol” and “Muppet Treasure Island,” before the company, which included the rights to “Sesame Street,” was eventually sold in 2000 to German media firm Em-TV for $680 million. Three years later, with Em-TV struggling with its new assets, the Hensons bought back the Muppets for $84 million, mostly in cash, before closing a deal with Michael Eisner, then chairman of the Walt Disney Co., in 2004 for the Muppets and children’s series “Bear in the Big Blue House.”Lisa says that, in a way, the Muppets were a model for how Disney subsequently acquired two other lucrative brands — Marvel and “Star Wars’” parent Lucasfilm. “It seems like a smart move for a creator to land there,” she adds. “We waited for so many years to close the deal with Disney for the Muppets to live forever.”

But there were some false starts when Kermit unpacked at the Mouse House. The Muppets were transferred to a division within the company called the Muppets Studios, staffed by just two employees. They starred in a handful of Web videos, but many Muppets fans fretted the characters were being overshadowed by other Disney properties until Segel, who said he wanted to take a shot at reviving the franchise, came along. Disney film exec Burr met with the multihyphenate to brainstorm ideas.

“It had been my opinion that in order to bring back the Muppets, we needed to add an element that was fresh and hip,” Burr says. “Jason was that.” She took the pitch to Dick Cook, the studio’s chairman at the time, and got his blessing. Segel agreed to take a pay cut, although he has since opted not to participate in the sequel. When asked by Variety about his former co-star, Kermit muses, “I think he wanted to move on and do other things,” adding diplomatically, “The door is always open for us to work together later on.”

On a recent afternoon, the Muppets are filming a YouTube cooking segment — another promotion for the film. Between takes, several of the puppeteers talk to Variety for a rare interview out of character. Along with Constantine’s Vogel, there are six other principal puppeteers who operate and perform voices for the Muppets — Steve Whitmire (Kermit, Beaker and Rizzo, to name a few), Eric Jacobson (Miss Piggy and Fozzie), Dave Goelz (Gonzo), Bill Barretta (the Swedish Chef and Rowlf the dog), David Rudman (Scooter) and Peter Linz (Walter). They are a tight-knit clan, most of them previously employed by the Jim Henson Co. Jacobson, who started as an intern, took over Oz’s characters, including Animal and Sam Eagle, when the puppeteer left that part of the job in 2000.

Jacobson says some people are still caught off guard that Miss Piggy is voiced by a man. “She’s one of the most famous drag acts in the business,” he quips. The Muppets have a policy, started by Henson himself, of making public appearances only when accompanied by their puppeteer. That means when Kermit does his press tour for “Muppets Most Wanted,” Whitmire is the only one who talks and operates him.

“We live their lives,” says Jacobson, who is more mild-mannered than his alter ego. “Everywhere Piggy is, I’m underneath. Everything she experiences, I experience.”

The new revival of the Muppets includes a Broadway show, which is in its early stages of development, but the Muppeteers are hoping for a new incarnation of “The Muppet Show” on TV.

“It’s just a matter of time,” Barretta says. “We’re excited to bring it back.” He has an idea for expanding the series. “I think it would be fun to go home and see their personal lives outside of work, and see things they get into. Simple everyday things, like ‘Curb Your Enthusiasm’ does.”

In a telephone interview, Brian Henson admits he had some reservations over the 2011 Disney reboot. “I felt like in the last movie, the Muppets were in one reality and the actors were in another, ” he says. But he calls “Most Wanted,” which he recently saw, a great Muppets movie. “I think my dad would be thrilled the Muppets are continuing,” he adds. “That’s a big deal.”

And if enough fans buy tickets, Kermit will stay green for many years to come.