Hurt was given three consecutive Academy Awards nominations for Best Actor for Kiss of the Spider Woman (1985), Children of a Lesser God (1986) and Broadcast News (1987.). He won for Kiss of the Spider Woman. Hurt earned his fourth Academy Award nomination for his supporting performance in A History of Violence (2005).
His film debut film role was in 1980, playing a scientist in the science fiction thriller Altered States, for which he received a Golden Globe nomination for New Star of the Year. He was the lawyer seduced by Kathleen Turner in Body Heat (1981). He was in the ensemble of The Big Chill (1983). He then appeared in the role of Arkady Renko in Gorky Park.
Other notable films include The Accidental Tourist, I Love You to Death, Lost in Space, Dune, AI Artificial Intelligence, Tuck Everlasting, The Good Shepherd, Mr. Brooks, Into the Wild, and Robin Hood.
Hurt was part of the Marvel Cinematic Universe films in as Thaddeus Ross, The Incredible Hulk, Captain America: Civil War, Avengers: Infinity War, Avengers: Endgame, and Black Widow, as well as the Marvel One-Shot The Consultant and the Disney+ animated series What If…?