Regarded as one of the foremost artists of the 20th century, Sondheim wrote the lyrics for West Side Story and composed Into the Woods among many others including Sweeney Todd and Company.
His friend and lawyer F Richard Pappas announced his death, which he described as “sudden”.
According to Pappas, Sondheim had spent the previous day having a Thanksgiving dinner with friends in Roxbury, Connecticut.
The worlds of theatre and film have been paying tribute.
Barbra Streisand wrote: “Thank the Lord that Sondheim lived to be 91-years-old so he had the time to write such wonderful music and GREAT lyrics! May he Rest In Peace .”
“I was just talking to someone a few nights ago about how much fun (and f***ing difficult) it is to sing Stephen Sondheim,” tweeted Anna Kendrick, who starred in the film adaptation of Into the Woods. “Performing his work has been among the greatest privileges of my career. A devastating loss.”
Cameron Mackintosh, the British theatre producer behind Les Miserables and Mary Poppins also paid tribute to Sondheim in a statement: “The theatre has lost one of its greatest geniuses and the world has lost one of its greatest and most original writers,” he said.
“Sadly, there is now a giant in the sky. But the brilliance of Stephen Sondheim will still be here as his legendary songs and shows will be performed for evermore. Goodbye old friend and thank you from all of us.”
Phantom of the Opera creator Andrew Lloyd Webber described Sondheim as a “musical theatre giant of our times, an inspiration not just to two but to three generations”.
He said Sondheim's contribution to theatre “will never be equalled”.
Josh Gad added: “Perhaps not since April 23rd of 1616 has theatre lost such a revolutionary voice. Thank you Mr Sondheim for your Demon Barber, some Night Music, a Sunday in the Park, Company, fun at a Forum, a trip Into the Woods and telling us a West Side Story. RIP. “
Broadway star Lea Salonga tweeted: “Rest In Peace, Stephen Sondheim, and thank you for your vast contributions to musical theater. We shall be singing your songs forever. Oh, my heart hurts…”
“Master. Legend. Icon. I was so privileged to join in 2 standing ovations last week at the opening of Company on Broadway,” actor Wilson Cruz tweeted. “Gratitude for the endless inspiration he provided to generations. He changed the world.”
Born in New York in 1930, Sondheim was tutored by the great composer Oscar Hammerstein and wrote his first musical at 15. He went on to have his first hit at just 27 with West Side Story which was a retelling of Romeo and Juliet set in 1950s New York City.
In a Broadway career that lasted over 60 years, Sondheim co-created other classics of the stage such as Gypsy, Sweeney Todd and Company.
Across his lengthy career, Sondheim won nine Tony Awards, an Academy Award, eight Grammys and and the Pulitzer Prize for Drama in 1985.
Sondheim was also awarded the Presidential Medal of Freedom by Barack Obama in 2015.