Born in the Bronx in New York City, Schuman played the violin and banjo as a child. As a young man he formed a dance band and wrote songs with his friend Frank Loesser. He went on to study at New York University's School of Commerce, but dropped out to study music instead, Roy Harris being among his teachers. Harris brought Schuman to the attention of the conductor Serge Koussevitzky, who championed many of his works.
In 1943 he won the inaugural Pulitzer Prize for Music for his cantata, A Free Song. In 1945, he became president of the Juilliard School of Music, founding the Juilliard String Quartet while there. He left in 1961 to take up directorship of the Lincoln Center.
Among Schuman's better known works are ten symphonies, a concerto for violin, the William Billings Overture and New England Triptych (both based on melodies by William Billings), the American Festival Overture, the ballet Undertow, and an opera about baseball, The Mighty Casey (based on Ernest L. Thayer's Casey at the Bat). He also arranged Charles Ives' organ piece Variations on "America" for orchestra, in which version it is better known.