All of this is part of the topic of perception, brilliantly and relevantly explored by writer/director Abbas Kiarostami in this film, a standout of the Cannes 2010 competition (and, from what I've seen, better than both the Palme d'Or and Gran Prix). This is a sensationally shot (by Luca Bigazzi) and composed film on nearly all levels; it has a spectacular mise-en-scene. It is also helped by Kiarostami's great ideas.
The film has been compared to "Before Sunset" by Richard Linklater, and the beginning and ending especially heavily reflect that. But they are birds of entirely different colors, though one could call "Certified Copy" "cynical 'Before Sunset'." Think of it this way: you know how when you see a film and you always want another that will give you a similar feeling? Then this will for lovers of the other film and its predecessor "Before Sunrise" (including myself).
Anyways, for some strange reason that's not I didn't entirely catch, the bulk of the film follows art critic/theorist Miller as he decides to take a day trip with Binoche out to Lucignano (an Italian middle ground for the French and the English), which is a famous place for artistic as well as matrimonial reasons. Along the way, he reveals himself as conceited and as somewhat undervaluing of and jaded about art, not being able to really see it in a human way anymore, not being able to feel "simple." I can relate. That's sometimes how I feel about watching films, sometimes not being able to connect to the popular feeling about a movie.
The two arrive and walk around before encountering the barista at the coffee shop, whose arrival (as people have said) triggers a shift toward the surreal, as the two begin to talk of their romantic past together. It's unclear whether or not this is supposed or true, although there are some clues along the way that could be construed as pointing one way or the other. I personally think this argument is beside the point. If, as another person has noted, they've been "certified," and if they are "married" long enough, it's as if they are. It doesn't matter if they really are. A good example of Kiarostami's wealth of food-for-thought. He brings the film to a sublime and perfect close which calls back an earlier topic of the film (framing the mundane) in a wonderful way.
In terms of the acting, it seemed as if Binoche faltered a bit in the "driving scene," though Kiarostami could have directed her that way on purpose. Otherwise, she's very good. Shimell is fabulous as well (in his transition from opera to film), doing extremely underrated work here. Going back to the subject of Kiarostami's direction, that's bit I'm least sure of, and I think it may stumble towards the beginning (with the lecture and the conversation between Binoche and her son, who's played by a so-so actor). But all into consideration, this is an astounding film, one that staggered and stimulated me. A-