You're absolutely right — prior to 2002's "The Bourne Identity," star Matt Damon's biggest films were all talky dramas such as "The Talented Mr. Ripley" (1999) and his breakout film, "Good Will Hunting" (1997). He was no action star, and he admitted as much himself at the time.
In a behind-the-scenes featurette for the DVD release, Damon said that he was given three months of intensive martial arts and weapons classes in advance of shooting: "It was like a summer school in assassin training."
The result was that "I ended up doing just about all of it," he concluded. Doing much of it himself was key to the film's realism.
"Movie audiences are so smart," Damon said, "they'll know if it's the actor doing it or if they're cutting away to a wide shot of some other guy who, you know, is much better at it."
Of course, that's not to say all the action is Damon himself. He had a personal stunt double (Laurent Boismaré, "Dialogue avec mon jardinier," 2007), and the film employed loads of other stunt performers for some of the more elaborate action scenes.
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