What is the fighting system Matt Damon uses in the Bourne movies?

I've looked into this too, because clearly Jason Bourne rocks! Here's what I've found, the fighting style Jason uses in the Bourne flicks is derived from an old Philippine martial arts style called Escrima and Pekiti Tirsia Kali. The Escrima being the Filipino martial arts that emphasizes stick and sword fighting. The Pekiti Tirsia Kali is more of a close quarters combat style; it is actually based upon a traditional Filipino dance. Well hopes this helps! Have fun learning it!

I've also seen him use a variation of Classic Good Looks in conjunction with George Clooney Clout.

yeah travistotz, I was just watching the extras in the Supremacy. The stunt coordinator called it Philippino Kali.