Roberts was born in New York City, the son of Norma, an animator, and CBS radio announcer Ken Roberts. He has a sister, Nancy, and is the cousin of late actor Everett Sloane. Roberts attended Northwestern University and made his Broadway debut in 1962, with a role in the play Something About a Soldier.
Roberts is best known for his collaborations with Woody Allen. In Annie Hall, he portrayed Alvy Singer's best friend Rob. Other Allen films and/or plays in which he has appeared include both the Broadway and film versions of Play It Again, Sam, Radio Days, Stardust Memories, Hannah and Her Sisters, A Midsummer Night's Sex Comedy and Woody Allen's segment for The Concert for New York City.
Roberts memorably portrayed the badgering Deputy Mayor Warren LaSalle in The Taking of Pelham One Two Three. He also appeared in the Sidney Lumet films Serpico and Just Tell Me What You Want.
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