Born in San Francisco, Anglim's original career goal was to be a veterinarian, but after he was asked to appear in a play, he switched to acting. He studied English literature at Yale, receiving a BA degree in 1973, and then spent a year in Connecticut at the Southbury Playhouse. His feature film debut was in 1973 in The All-American Boy, and his first television appearance was in 1976 in the PBS The Adams Chronicles.
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. In 1982, Anglim appeared in the NBC television version of the play, and earned Best Actor nominations for the performance.
Anglim also maintains a cattle farm in Tennessee, and in 1992, founded The Lewis County Children's Fund, to help children in the area.
All of the following were for his performance in The Elephant Man:
