Born in the U.S. state of Virginia and educated at Princeton University B.A. summa cum laude in 1946, with postgraduate study at the University of Rome in 1949. Weaver was an ambulance driver in Italy during World War II for the American Field Service, and lived primarily in Italy after the end of the war. Through his friendships with Elsa Morante, Alberto Moravia and others, Weaver met many of Italy's leading authors and intellectuals in Rome in the late 1940s and early 1950s; he paid tribute to them in his anthology Open City (1999).
Most recently, Weaver was a professor of literature at Bard College in New York, and a Bard Center Fellow. He received honorary degrees from the University of Leicester in the United Kingdom and Trinity College in Connecticut. According to translator Geoffrey Brock, Weaver was too ill to translate Umberto Eco's 2005 novel, The Mysterious Flame of Queen Loana (La misteriosa fiamma della regina Loana 2004).