Tudor was born in Philadelphia, Pennsylvania. He studied piano with Stefan Wolpe and became known as one of the leading performers of avant garde piano music. He gave the first American performance of the Piano Sonata No. 2 by Pierre Boulez in 1950, and a European tour in 1954 greatly enhanced his reputation. Karlheinz Stockhausen dedicated his Klavierstück VI (1955) to Tudor. Tudor also gave early performances of works by Morton Feldman and La Monte Young.
The composer with which Tudor is particularly associated is John Cage. He gave the premiere of Cage's Music of Changes, Concerto For Piano and Orchestra and the notorious 4' 33". Cage said that many of his pieces were written either specifically for Tudor to perform or with him in mind. The two worked closely together on many of Cage's pieces, both works for piano and electronic pieces. Upon Cage's death in 1992, Tudor took over as music director of the Merce Cunningham Dance Company.
After a stint teaching at Darmstadt from 1956 to 1961, Tudor began to wind up his activities as a pianist to concentrate on composing. He wrote mostly electronic works, many commissioned by Merce Cunningham, and often with elaborate lighting and stage designs. One piece, Reunion (1968), written jointly with Lowell Cross features a chess games, where each move triggers a lighting effect or projection. At the premiere, the game was played between John Cage and Marcel Duchamp.
Tudor died in Tompkins Cove, New York.