His death, from an apparent heart attack, was confirmed by his son Mark Donen.

A former Broadway dancer, Mr. Donen (pronounced DAH-nen) first drew notice as a wunderkind whose innovations in movie choreography literally animated film in new ways. At 20, he conceived and directed Kelly’s timelessly endearing dance duet with Jerry the cartoon mouse — a pioneering concept — in “Anchors Aweigh” (1945).

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Mr. Donen’s alliance with Kelly resulted in some of the finest work either of them achieved on-screen. They co-directed three films: “On the Town” (1949), “It’s Always Fair Weather” (1955) and, towering above Mr. Donen’s dozens of credits, “Singin’ in the Rain” (1952).

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The musical starred Kelly, Debbie Reynolds and Donald O’Connor and lampooned silent-era Hollywood studios during their bumpy transition to the “talkies.” It ranks fifth on the American Film Institute’s list of the greatest American movies of all time. Scores of influential critics have lauded its witty script by Betty Comden and Adolph Green, brisk pacing and imaginative dance numbers.

Under Mr. Donen’s direction, the title song sequence climaxed with the camera swooping back into the air as Kelly spins in circles, riding his umbrella like a sail. The image is considered one of the most extraordinary explosions of joy ever captured on film.

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In an interview, film scholar David Thomson called Mr. Donen “one of the handful of absolutely vital contributors to the American movie musical.” He achieved much of his best work in the collaborative ambiance of the studio system of the 1940s and 1950s.

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He teamed with choreographers Bob Fosse and Michael Kidd, as well as esteemed producers Arthur Freed and Roger Edens. But he forged his most important working relationship with Kelly, although the Kelly-Donen partnership eventually shattered over dueling egos. “If you substitute the word ‘fight’ for ‘co-direct,’ then you have it,” Mr. Donen once said.

Mr. Donen had strong ideas about what worked in film, but without Kelly’s star clout, he did not always see them realized. Film historian and critic Andrew Sarris saw “intermittent flashes of inspiration” in Mr. Donen’s non-Kelly musicals such as “Royal Wedding” (1951) and “Funny Face” (1957), both with Astaire, and his movie versions of the Broadway shows “The Pajama Game” (1957) and “Damn Yankees!” (1958).

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In making “Seven Brides for Seven Brothers” (1954), a musical about macho backwoodsmen on the American frontier, Mr. Donen and Kidd persuaded producers who wanted name actors in several leading roles to cast professionally trained ballet dancers instead.

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Mr. Donen said producers feared that the dancers would seem unmanly to a mass audience. But the result was the unforgettable and richly acrobatic barn dance sequence, which featured men cartwheeling on planks and wielding axes. It was, a Time magazine critic wrote, “a high old roister-doister of a show.”

Mr. Donen advocated liberating the camera from the studio set — an expensive and extremely rare practice. He was disappointed to have lost that battle in “Seven Brides,” when he was forced to use a studio backdrop that looked artificial then and seems severely dated now.

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He had better luck, with Kelly’s pull, when it came to filming the musical “On the Town” in New York. The opening sequence, aided by Leonard Bernstein’s brassy score, was bursting with dynamic energy and starred Kelly, Frank Sinatra and Jules Munshin as sailors on shore leave.

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“It was important, in ‘On the Town,’ to be on the town,” Mr. Donen once quipped.

Location shooting at the Louvre in Paris also proved effective in “Funny Face,” starring Astaire as a worldly photographer and Audrey Hepburn as a young mouse he transforms into a Givenchy-clad fashion goddess.

In a scene that embodies Mr. Donen at his most effervescent, Hepburn rushes down the stairs in a pose that resembles the looming marble sculpture “Winged Victory of Samothrace.”

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“I can’t stop,” she tells Astaire to swelling Gershwin music. “Take the picture!”

Stanley Donen was born April 13, 1924, to a Jewish family in Columbia, S.C., where his father managed a dress shop. To escape anti-Semitic taunting at school, he took refuge in Astaire and Ginger Rogers musicals that offered a “fantasy world where everything seemed to be happy, comfortable, easy and supported.”

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He trained as a dancer and, at 16, won a role in the Broadway chorus line of “Pal Joey” (1940), a hit musical starring Kelly. The next year, Mr. Donen became an assistant to Kelly, who was choreographing the musical “Best Foot Forward” in New York. Their partnership deepened in Hollywood.

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Mr. Donen’s breakthrough as a movie choreographer was the “Alter Ego” sequence for the 1944 musical “Cover Girl.” He used double exposure — a novel technique for Hollywood movies — to simulate Kelly dancing with a ghostlike figure of himself on a lonely street block.

“The sequence occurred to me because two people dancing together is more powerful, precise, and fun than one person dancing alone,” Mr. Donen told his biographer, Stephen M. Silverman. “Then I thought: What if those two people are the same person? So I went to Gene and asked him, ‘How’d you like a number where you dance with yourself?’ Well, the idea appealed to him on a number of levels.”

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Veteran director Charles Vidor did not think Mr. Donen’s idea was possible. But the young choreographer achieved the effect by shooting the precise camera movements and dance steps twice. Kelly’s perfectionism — and his vouching for Mr. Donen — were crucial to the success.

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The actor described being drawn to Mr. Donen, who was 11 years his junior, because he recognized someone who matched his energy and ambition. But Kelly’s demanding ways and temper eventually led to a falling-out. It didn’t help that the first of Mr. Donen’s five wives, dancer Jeanne Coyne, subsequently married Kelly. Kelly died in 1996.

“I’m grateful to him,” Mr. Donen remarked in a 1992 interview, “but I paid back the debt 10 times over. And he got his money’s worth out of me.”

Mr. Donen’s later marriages to actress Marion Marshall, socialite Adelle Beatty, actress Yvette Mimieux and saleswoman Pam Braden ended in divorce. For several years, he was the companion of writer, actress and filmmaker Elaine May.

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Survivors include a son from his second marriage, Joshua Donen, a producer of the Netflix series “House of Cards” and the 2014 film “Gone Girl,” of Los Angeles; a son from his third marriage, Mark Donen of Paris; a sister; and three grandchildren. A son from his second marriage, movie visual effects specialist Peter Donen, died in 2003.

Mr. Donen made his solo directing debut with “Royal Wedding,” a film best remembered for Astaire’s tap dance around a hotel room, from floor to wall to ceiling and back again.

To achieve the dizzying effect, Mr. Donen attached his camera to the base of a rotating cylindrical set. The cylinder would turn as Astaire danced, leaving the impression of a man defying the laws of gravity. To add to the illusion, background “draperies” were made of wood, and Astaire’s jacket was sewn into a chair that was anchored to the floor.

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Mr. Donen repeated his wizardry in 1986 when he directed a music video of Lionel Richie’s best-selling single “Dancing on the Ceiling.”

As big-budget musicals went into decline in the late 1950s, Mr. Donen moved to London and made several sleek European-set stories. The most notable was the popular romantic-comic thriller “Charade” (1963), which featured a sexy slapstick sequence in which Cary Grant takes a shower fully clothed as Hepburn watches.

In 1967, Mr. Donen made the popular hit “Bedazzled” featuring the comedy team of Dudley Moore and Peter Cook. That same year, the director tried a nonlinear storytelling approach in “Two for the Road,” a fractured romance starring Hepburn and Albert Finney. The film failed commercially but has since attracted a devoted following for its avant-garde style.

Mr. Donen dismissed many of the movies he made in the 1950s and 1960s as well-intentioned trifles best forgotten. They included “Staircase” (1969), with Rex Harrison and Richard Burton as an aging gay couple.

He ended his feature-film career on a sour note with “Blame It on Rio” (1984), a dirty-old-man farce starring Michael Caine. An exception to his poorly received later film work was “Movie Movie” (1978), a sendup of a 1930s double feature that included a fake trailer for a World War I flying-ace drama (“War at Its Best!”). The movie, starring George C. Scott, was a deeply felt and riotously silly homage to Mr. Donen’s earliest memories at the movies.

He also produced a well-received 1986 Academy Award telecast that included Robin Williams as a host.

In 1998, Mr. Donen received an honorary Oscar for lifetime achievement “in appreciation of a body of work marked by grace, elegance, wit and visual innovation.” Director Martin Scorsese presented the award to Mr. Donen, who celebrated the moment by tap dancing to the Irving Berlin song “Cheek to Cheek” with the Oscar statuette.

“Singin’ in the Rain” influenced later films, including the Oscar-winning “The Artist” (2011) and Stanley Kubrick’s “A Clockwork Orange” (1971), which borrowed the song for nightmarish effect.

For all its later acclaim, “Singin’ in the Rain” drew favorable but not ecstatic reviews when it was released. Mr. Donen said it was overshadowed by director Vincente Minnelli’s “An American in Paris” (1951), which also starred Kelly and won the Oscar for best picture. “Singin’ in the Rain” was nominated in two minor categories the next year but lost.

“We were ignored,” Mr. Donen told his biographer. “Not that it’s such a big to-do. The year of ‘Singin’ in the Rain,’ the best picture went to ‘The Greatest Show on Earth,’ one of the worst movies ever made.”