Note: I have only seen the film version, but from what I can tell, it's faithful to the comic. Therefore, a lot of my compliments to the film are probably compliments to the original comic by Satrapi.
Persepolis is not like many other animated movies. It does not take place in a fantastic world, unless you consider the one we live in today fantastic. Its characters are drawn far more cartoonishly than in other traditionally animated films, yet they are far more real. There is no climax, only a sad goodbye. The film owes its strangeness to the origin of the story, the life of its author Marjane.
The animation is one of my favorite aspects. It imitates the look of the comic, with simply illustrated black-and-white characters that look demented and silly as easily as they look somber and grounded in reality. The entire thing flows very well, and there is clever use of scene transitions as framing devices. One thing I was very surprised by was how recognizable the characters were; while the faces are simple and 2-dimensional, most of the film is in black-and-white, and the women often wear headscarfs that make them look more similar, I never had trouble telling who was who.
The story is not a traditional one with a three act structure. Instead, it is a story of life that imitates all triumphs and pitfalls of our everyday existence. Depression, love, loss, bravery, selfishness, even simple pondering over the role of God... Marjane encounters all of these because that's what normal people deal with, and that's why her journey is so interesting to me. It's all real, in more ways than one.
Finally, there's the political side of this. It can be very blunt, but it's blunt for a reason; when people live under oppressive regimes (Like the dictatorship my father lived under), they feel the effects all the time. It's difficult for us who have not lived in those places to comprehend. It's also sometimes a good thing to get knocked out of our comfort zone.
Hopefully, we'll see more animated works like Persepolis in the future; works that deal with life and its troubles instead of just artificially forced-in dramatic conflicts, works that question our