Shaw grew up in New Haven, Connecticut, and played the alto saxophone in a dance band there as a teenager. He moved to Cleveland in 1926 and to New York in 1929; there he played in Harlem jam sessions with pianist Willie "the Lion" Smith. In the early 1930s, he worked as a session musician.
His first band, formed in 1936, included a rhythm section and string quartet; in 1937, he formed a swing band, and scored his first big hit a year later with Begin the Beguine. Over the next several years, he had several more hits, including Frenesi and Summit Ridge Drive, with various ensembles. But he found the pressures of fame unbearable‹gossip columnists were fascinated by his movie-star good looks and his eight marriages‹and took a series of sabbaticals from performing.
In 1942, Shaw enlisted in the Navy; he formed a new band and performed for servicemen throughout the Pacific theater of the war. After the war, he played with small groups and with two different big bands, and spent a year studying classical music. He retired from music altogether in 1954, and spent much of the second half of his life writing fiction. In 1983, his band was reformed under the leadership of clarinetist Dick Johnson, and he occasionally conducted the new group.
A perfectionist and an intensely private man, Shaw could be difficult to work with. Late in life, he told the editors of Who's Who that he wanted his obituary to read, "He did the best he could with the material at hand." Later, he told a lecture audience that he had come up with a shorter, more elegant version: "Go away."