His feature film debut was in The All-American Boy (1973), and his first television appearance was in the PBS The Adams Chronicles (1976). In 1979, while still an unknown, Anglim optioned the London play The Elephant Man, and debuted off-Broadway at St. Peter's Church. It later moved to The Booth Theater on Broadway in 1979, and earned several awards. He performed as Macbeth on Broadway in 1981 (in which he was replaced by Kelsey Grammer) and again on TV in 1982. In 1982, Anglim appeared in the ABC television version of The Elephant Man, and earned an Emmy nomination for Best Actor for the performance.