He died of complications from esophageal cancer, according to a statement from his publicist.
Demme won consistent acclaim as the director of such diverse movies as the Talking Heads concert film "Stop Making Sense"; "Philadelphia," the 1993 drama starring Tom Hanks as a lawyer battling AIDS; and "Beloved," the 1998 Oprah Winfrey movie based on Toni Morrison's bestseller about a 19th century slave haunted by the ghost of her daughter.
But it's "Silence of the Lambs" for which he was best known. The 1991 movie, based on Thomas Harris' novel, swept the Academy Awards the following year, winning Oscars for best picture and its two stars, Jodie Foster and Anthony Hopkins, whose Hannibal Lecter character became a cultural icon. Demme also won the Oscar for best director.
The movie was a box-office hit, spawned several sequels and a TV series, and chilled viewers with its portrait of Lecter, a refined but cannibalistic serial killer who bragged about eating a victim's liver "with some fava beans and a nice Chianti."
Over a career that spanned four decades Demme directed an eclectic mix of films, including the Michelle Pfeiffer comedy "Married to the Mob," the Melanie Griffith-Jeff Daniels road-trip adventure "Something Wild," a remake of political thriller "The Manchurian Candidate" with Denzel Washington, and the Anne Hathaway indie drama "Rachel Getting Married."
He also was a big lover of music, filling his movies with songs and directing documentaries or videos for such artists as Neil Young, the Pretenders, Bruce Springsteen and Justin Timberlake.
"Sadly, I can confirm that Jonathan passed away early this morning in his Manhattan apartment, surrounded by his wife, Joanne Howard, and three children," the publicist said. There will be a private family funeral.
Demme was known for a compassionate approach to his characters and his use of dramatic close-ups, most effectively in "Silence of the Lambs'" tense exchanges between Hopkins ("quid pro quo, Clarice") and Foster's young FBI trainee.
By Wednesday afternoon tributes to Demme were pouring out of Hollywood.
"I am heartbroken to lose a friend, a mentor, a guy so singular and dynamic you'd have to design a hurricane to contain him," Jodie Foster said in a statement to Rolling Stone. "Jonathan was as quirky as his comedies and as deep as his dramas. He was pure energy; the unstoppable cheerleader for anyone creative."
"Jonathan taught us how big a heart a person can have, and how it will guide how we live and what we do for a living," Tom Hanks, who won an Oscar for "Philadelphia," said in a statement to Entertainment Weekly. "He was the grandest of men."
Unlike many directors of his generation, Demme didn't go to film school. He broke into the industry through a friendship with cult filmmaker Roger Corman, who hired him to write and direct such low-budget genre movies as "Caged Heat," about inmates in a women's prison.
Demme's critical breakthrough was "Melvin and Howard," a 1980 film about a much-disputed encounter between gas station owner Melvin Dummar and reclusive billionaire Howard Hughes.
Other films included "Swing Shift," a period romance with Goldie Hawn; "Swimming to Cambodia," a Spalding Gray monologue; and the documentary "Jimmy Carter Man from Plains," about the former president.
His final film was 2015's "Ricki and the Flash," starring Meryl Streep as an aging rocker.
Jonathan Demme: Silence of the Lambs stars Foster and Hopkins pay tribute
Jodie Foster and Sir Anthony Hopkins have paid tribute to the late Jonathan Demme, who directed them in Oscar-winning film The Silence of the Lambs.
Foster said his death had left her "heartbroken", while Sir Anthony remembered him as "one of the best".
Actress Meryl Streep called his death "a big loss to the caring world", while Talking Heads' David Byrne said he had been "hugely inspirational".
Demme died on Wednesday, aged 73, from complications from oesophageal cancer.
He won an Academy Award for directing The Silence of the Lambs, which also won Oscars for best picture, best screenplay and for Foster and Hopkins' performances.
Foster, who played FBI trainee Clarice Starling in the 1991 thriller, remembered Demme as "a friend, a mentor, a guy so singular and dynamic you'd have to design a hurricane to contain him.
"Jonathan was as quirky as his comedies and as deep as his dramas. He was pure energy, the unstoppable cheerleader for anyone creative.
"Just as passionate about music as he was about art, he was and will always be a champion of the soul," the actress and director continued.
"JD, most beloved, something wild, brother of love, director of the lambs. Love that guy. Love him so much."
"I am really shocked and very sad to hear about Jonathan's passing," said Sir Anthony Hopkins in his own tribute to the film-maker.
"He was one of the best, and a really nice guy as well who had such a great spirit. Every day being with him was a high five."
In her tribute, Streep described Demme as "a big-hearted, big tent, compassionate man - in full embrace in his life of people in need - and of the potential of art, music, poetry and film to fill that need."
The three-time Oscar winner worked with the director on his 2004 remake of The Manchurian Candidate and in 2015's Ricki and the Flash.
In a lengthy tribute on his website, Byrne said Demme's films had been "filled with so much passion and love" and that his world view had been "open, warm, animated and energetic".
"He was directing TV episodes even this year, when he was in remission," the singer and musician revealed.
The Talking Heads frontman worked with the director on concert film Stop Making Sense and also wrote music for his films Something Wild and Married to the Mob.
Further tributes have been paid by Reese Witherspoon, who remembered Demme as "a great film-maker and a lovely man", and Anne Hathaway, who called him her "favourite rebel angel".
Tim Robbins remembered him as "a lovely man, a great director and a kind soul", while Robbins' former partner Susan Sarandon said she was "devastated".
"Words just aren't enough with heartache like this," wrote pop star Justin Timberlake, who worked with Demme on a concert film last year.
"The time I spent with you away from a camera and a stage made a better human out of me. You are truly irreplaceable."
Demme's publicist, Leslee Dart, confirmed on Wednesday that the director died on Wednesday morning in his Manhattan apartment.
A private family funeral will be held for the director, who is survived by his wife, Joanne Howard, and his children Ramona, Brooklyn and Jos.