The two are friendly in real life—the actor even attended the Jimmy Kimmel Live! host's 2013 wedding. But on TV, they are arch enemies.
It all began in the early times of the show, which premiered in 2003. Kimmel closed his Jimmy Kimmel Live! shows with, "Apologies to Matt Damon, we ran out of time"—even though he wasn't scheduled to appear. It soon became a recurring gag and led to an epic mock celebrity feud that continues to this day.