Demme won the best director Oscar in 1992 for The Silence of the Lambs
His publicist confirmed he died from complications from oesophageal cancer.
Born in 1944, Demme’s other features included Philadelphia, Something Wild and the Talking Heads documentary Stop Making Sense.
Tom Hanks, who won an Oscar for his performance in Philadelphia, told the Press Association Demme was “the grandest of men”.
He said: “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.”
Demme’s own Oscar was for best director for The Silence of the Lambs in 1992.
The second film to feature serial killer Hannibal Lecter, it is one of only three films to win the so-called “big five” Oscars.
As well as best director, the 1991 film was named best picture, won a screenplay prize and saw both of its lead actors honoured.
In recent years he worked with Anne Hathaway on Rachel Getting Married and directed Meryl Streep in both Ricki and the Flash and his 2004 remake of The Manchurian Candidate.
His most recent film, Justin Timberlake + the Tennessee Kids, showed Timberlake in concert in 2015.
Tributes flowed in from the film world:
British actress Thandie Newton, who worked with him on Beloved and The Truth About Charlie, said she was “deeply saddened” by his passing
Demme also steered Mary Steenburgen to a best supporting actress Oscar for his 1980 film Melvin and Howard.
Actor-turned-director Ron Howard remembered Demme as “a great artist, humanitarian, activist & a warm encouraging colleague”.
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