Wolpe was born in Berlin. He studied composition under Hans Schreker and was also a pupil of Ferruccio Busoni. He also studied at the Bauhaus and met some of the dadaists, setting Kurt Schwitters' poem Anna Blume to music.
The music Wolpe was writing was atonal, using Arnold Schoenberg's twelve tone technique. However, possibly influenced by Paul Hindemith's concept of Gebrauchsmusik (music which serves a social function) and, as an avid socialist, he wrote a number of pieces for worker's unions and communist theatre groups. For these he made his style more accessible, incorporating elements of jazz and popular music. His songs became popular, rivalling those of Hanns Eisler.
When the Nazis came to power in Germany, Wolpe fled the country, passing through Romania and Russia en route to Austria, where he met and studied with Anton Webern. He later moved to Palestine where he wrote simple songs for the kibbutzim. The music he was writing for concert performance, however, remained complex and atonal.
In 1938, Wolpe moved to New York City in the United States of America. There he assocaited with the abstract expressionist painters. From 1952 to 1956 he was director of music at Black Mountain College, and he also lectured at the summer schools in Darmstadt in Germany. His pupils included Morton Feldman, David Tudor and Charles Wuorinen.
His works from this time sometimes used the twelve tone technique, were sometimes diatonic, were sometimes based on the Arabic scales he had heard in Palestine and sometimes employed some other method of tonal organisation. His work was radical, but avoided the pointillism of composers such as Pierre Boulez, instead employing more conventionally expressive gestures.