If you take a step back from “The Bourne Ultimatum,” you might notice that this movie has virtually no plot, contains nothing that we didn’t already know, and really is just one big long chase scene.

But that’s the thing about “The Bourne Ultimatum”: You can’t take a step back from this movie. It grabs you hard in its opening moments and doesn’t let go until the satisfying fade-out.

This is part three of a spy trilogy nominally based on books by Robert Ludlum. In “The Bourne Identity,” we met a CIA spy called Jason Bourne (Matt Damon), an amnesiac troubled by what he’d done in the past – a past he couldn’t remember.

“The Bourne Supremacy” followed, with director Doug Liman replaced by Paul Greengrass. Greengrass is back for this installment, which uses his trademark jittery handheld camera throughout (he did the excellent “United 93”).

Like the previous movies, “Ultimatum” has a cynical, unsentimental view of the spy business. In this film, there’s no enemy except the CIA operatives who set Bourne up to be a killing machine in the first place. He seeks an answer to the mystery of his identity, and justice for a character killed in “The Bourne Supremacy.”

Two characters from the previous films, CIA agents played by Joan Allen and Julia Stiles, return for significant roles in this one; this is the rare sequel that actually expands the meaning of its supporting characters.

The globe-hopping is back, too, with Bourne bopping from Madrid to London to Tangier and elsewhere in his search. He’s being tracked by a lethal CIA honcho (David Strathairn) for whom the phrase “terminate with extreme prejudice” is insufficiently vague.

The film is arranged around a handful of elaborate, incredibly detailed chase sequences. The first takes Bourne and an English journalist (Paddy Considine) through London’s Waterloo Station, and it’s so taut (and logical) you figure the film can’t possibly top itself.

But it does, repeatedly. At times “Ultimatum” threatens to become a technical exercise: How can Greengrass stage a foot chase through the crowded streets of Tangier and make it feel fresh? But he keeps making it feel fresh, and you keep leaning forward in your seat.

This nervous style, full of jump cuts and shaky motion, is something I generally don’t like. So I’m even more impressed that Greengrass pulls it off here.

Through it all, Matt Damon maintains his brutally hard composure, which asks no sympathy from the audience. It’s tricky to bottle everything up the way Damon does with this cleaned-out character, but the movie would collapse without his spooky self-control. And since there’s nowhere else for Bourne to go, this “Ultimatum” should indeed be final.

Matt Damon stars in “The Bourne Ultimatum.”