His orchestra manager, Will Curtis, said Shaw had been ill for some time, although the precise cause of death was not known.
Shaw, who at his peak ranked alongside masters of swing such as Benny Goodman, Tommy Dorsey and Glenn Miller, became famous at 28 with the release of his band's recording of Cole Porter's Begin the Beguine. Initially intended as a B-side, it headed the charts for six weeks in 1938. At the height of his career Shaw was earning $30,000 a week - an enormous sum during the Depression era. His other hits included Lady Be Good, Indian Love Call and Frenesi.
Shaw worked with some of the great jazz legends, including Buddy Rich, Mel Torme and Gordon Jenkins and Billie Holiday - a significant act at a time when most bandleaders would not hire African Americans.
He stopped playing the clarinet in 1954, claiming that since he could not reach the level he wished to attain he should never play it again.
"I am compulsive. I sought perfection. I was constantly miserable. I was seeking a constantly receding horizon. So I quit," he said in an interview with Reuters in 1985.
"It was like cutting off an arm that had gangrene. I had to cut it off to live. I'd be dead if I didn't stop. The better I got, the higher I aimed. People loved what I did, but I had grown past it. I got to the point where I was walking in my own footsteps."
Shaw withdrew from music but not from public view, appearing instead on television gameshows.
He also wrote an autobiography and a novel, travelled and appeared on the lecture circuit.
In the early 80s, however, he made a comeback, with a band that bore his name and played his music but had a different clarinetist - Dick Johnson - who lead the orchestra and played the solos.
Shaw, who had eight wives throughout his life, often confessed that he was a difficult man to get along with.
His longest marriage was to his last wife, Evelyn Keyes, who played the middle of the three O'Hara sisters in Gone With the Wind. The marriage lasted 28 years although for much of it they lived separately.
"I like her very much and she likes me, but we've found it about impossible to live together," he said in an interview in 1973.