Crumb was born in Charleston, West Virginia, and began to compose at an early age. In 1947 he studied at the National Music Camp in Interlochen, Michigan. He majored in music at the Mason College of Music and Fine Arts (subsequently subsumed into the University of Charleston), where he received his bachelor's degree in 1950. He obtained his M.Mus. at the University of Illinois at Urbana–Champaign in 1952 and then briefly studied as a Fulbright fellow at the Hochschule für Musik in Berlin before returning to the United States to study at the University of Michigan, from which he received a D.M.A. in 1959.
Crumb has earned his living primarily from teaching. His first teaching job was at a college in Virginia, before he became professor of piano and composition at the University of Colorado in 1958. In 1965 he began a long association with the University of Pennsylvania, becoming Annenberg Professor of the Humanities in 1983. Some of his most prominent students include Margaret Brouwer, Uri Caine, Christopher Rouse, Osvaldo Golijov, Jennifer Higdon, Cynthia Cozette Lee, Yen Lu,
Thomas Meadowcroft , Ofer Ben-Amots, Robert Carl, and Gerald Levinson.
Crumb retired from teaching in 1997, though in early 2002 he was appointed with David Burge to a joint residency at Arizona State University. He has continued to compose.
Crumb has been the recipient of a number of awards, including a 1968 Pulitzer Prize for Music for his orchestral work Echoes of Time and the River and a 2000 Grammy Award for Best Contemporary Composition for his work Star-Child .
Crumb's son, David Crumb, is a successful composer and, since 1997, assistant professor at the University of Oregon. George Crumb's daughter, Ann Crumb, is a successful actress and singer. She recorded his Three Early Songs for the CD George Crumb 70th Birthday Album (1999), and has also performed his Unto the Hills (2001).