Born in Los Angeles, California, he was the son of Darryl Zanuck, the famed head of Twentieth-Century Fox studios. While studying at Stanford University, Richard began his career in the film industry working for the Twentieth-Century Fox story department. In 1959, Zanuck got his first shot at producing when his father installed him as the producer of the film Compulsion. In the 1960's Zanuck became the president of his father's studio; one year of his tenure, 1967, is chronicled in the John Gregory Dunne novel "The Studio". Richard was later fired by his father and joined Warner Brothers, although Zanuck was eventually fired from there too.

In 1972, Zanuck joined up with David Brown to form an independent production company called Zanuck/Brown Productions. With help from Universal Pictures, the pair produced two of Steven Spielberg's early films, The Sugarland Express (1974) and Jaws (1975). They went on to produce such box office hits as Cocoon (1985) and Driving Miss Daisy (1989) before dissolving their partnership in 1988. Nevertheless, they were jointly awarded The Irving G. Thalberg Memorial Award by the Academy of Motion Picture Arts and Sciences in 1990.

In 2000, Zanuck and his third wife Lili Fini Zanuck co-produced the 72nd Academy Awards ceremony.