Trained by his parents (a cellist and a pianist) and at the Moscow Conservatory (1943–48), Rostropovich became professor of cello at the conservatory in 1956. He began touring abroad in the 1950s. He also performed as a pianist in recitals with his wife, the soprano Galina Vishnevskaya, and in 1968 he made his debut as a conductor. When in 1970 Rostropovich made clear his support of the dissident Soviet writer Aleksandr Solzhenitsyn (Solzhenitsyn, Aleksandr Isayevich), the government sharply curtailed his ability to travel. In 1974, however, he and his wife were permitted to leave the country, and in 1975 they announced their decision not to return to the Soviet Union. In 1977 Rostropovich became music director of the National Symphony Orchestra in Washington, D.C., a post he held until 1994. The Soviet government deprived the couple of their citizenship in 1978 but reversed that decision in 1990.

Although sometimes criticized for occasional over-romanticism, Rostropovich was admired for his keen musicianship, both in contemporary works and in the established concert repertoire. His exploitation of the tonal resources of the cello was considered exceptional. Composers who wrote works for him include Aram Khachaturian (Khachaturian, Aram), Sergey Prokofiev (Prokofiev, Sergey), Dmitry Shostakovich (Shostakovich, Dmitry), Benjamin Britten (Britten, Benjamin), and Witold Lutosławski (Lutosławski, Witold). The recipient of numerous awards, Rostropovich was given the Presidential Medal of Freedom in 1987.