- In 2002, received Toronto Film Critics Association Award for Best Performance, Male
- In 2001, received ShoWest Award for Decade of Achievement
- In 1998, received MTV Movie Award for Best Onscreen Duo
- In 1997, received MTV Movie Award for Best Onscreen Duo
- In 1996, received Academy Award for Best Actor
- In 1996, received National Society of Film Critics Award for Best Actor
- In 1996, received Screen Actors Guild Award for Outstanding Performance by a Male Actor in Leading Role
- In 1996, received Toronto Film Critics Award for Lifetime Achievement
- In 1996, received Chicago Film Critics Award for Best Actor
- In 1996, received Golden Globe Award for Best Actor in a Motion Picture (Drama)
- In 1995, received National Board of Review Award for Best Actor
- In 1995, received Los Angeles Film Critics Association Award for Best Actor
- In 1995, received San Sebastian Film Festival Award for Best Actor
- In 1995, received Boston Society of Film Critics Award for Best Actor
- In 1995, received New York Film Critics Circle Award for Best Actor
- In 1995, received Sundance Film Festival Award for Independent Vision
