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Caution: potential spoilers.
As usual, there are plenty of different categories where this review might fit. The movie's slightly weird, but not enough to put it under "weird." It's certainly artistic, but I still wouldn't call it an "art film." There are definite "coming of age" elements. Some of the fighting borders on "martial arts." I can't call it a "spy flick," in spite of the backstory and driving premise of the movie having to do with spies. There's a bit of a "science fiction" element to it, but I won't say how. (It would be a bit spoilery, even if I think it's totally predictable.) And I suppose there are any number of other elements, but I think the most apt description would be "action/adventure," so that's how I'm categorizing it. (Edit: I later moved it to "badass.")
It starts with a 16-year-old girl named Hanna (Saoirse Ronan) hunting and killing a reindeer or whatever. She then gets into a fight with a man who turns out to be her father, Erik Heller (played by Eric Bana). They live in a forest in Finland (so I learned online; I never caught any mention in the movie of exactly where this wintery forest was, though someone probably said it). Anyway, Erik has been training her all her life, while they've been in hiding, living a simple existence. So, Hanna is strong, fast, agile, excellent at hand-to-hand combat as well as using both bows & arrows and guns. She also has a great deal of book-learning, so that she can recite facts in an encyclopedic fashion (which apparently is meant to help her understand the outside world, which she's never seen, though actually it makes her seem quite odd). She also speaks several languages. Predictably, she eventually decides she wants to go out and see the world (in part because she's never heard music, but only knows its technical definition). So, Erik presents her with a beacon that has a switch she can flip if she believes she is truly ready. That will allow someone named Marissa Wiegler (Cate Blanchett) to find them. And Erik warns Hanna that if Marissa does find Hanna, she won't stop until either she or Hanna is dead. Of course, it isn't long before Hanna decides to flip the switch. (I don't really get the point of that; it seems like it would've been simpler for Erik and Hanna to just leave their cabin and go out into the world without purposely attracting the attention of this deadly antagonist. I suppose you can simply accept it as an implausible plot device, or you could say that Erik wanted Marissa dead, and maybe was using this as a final test to prove if Hanna is truly ready to live in the real world. Or it could be that he knew sooner or later Marissa would find them anyway, and wanted it to happen at a time of his choosing, so Hanna wouldn't have to spend the rest of her life looking over her shoulder. Still, it can't help feeling like a plot device, to me.)
Anyway... Hanna flips the switch. Then she cuts her father's hair, he puts on some some civilized clothes, and he leaves her alone. It's not long before a CIA strike team shows up, sent by Marissa to find Erik. Instead, they just find Hanna, who takes a couple of them down before being captured. The scene then changes to her being held in some top-secret facility, where she's being questioned by a psychiatrist, and observed via video feed by Marissa and other CIA agents. She specifically asks to see Marissa, and a double is sent in claiming to be her, while the real Marissa continues to watch. It's not long before Hanna kills the fake (she'll later send a post card to Erik telling him "the witch is dead"), and proceeds to kill a great number of guards while making her escape.
She finds herself in the desert, in Morocco, where she soon makes the acquaintance of a British teenage girl named Sophie, who's apparently on vacation with her little brother Miles, and their parents, Rachel and Sebastian, traveling in an RV. Hanna ends up traveling with the family (sometimes without their knowledge, sometimes by invitation). Miles seems to have a bit of a crush on Hanna, though the movie spends more time on the relationship between Hanna and Sophie, who become friends (and possibly more). They travel to Spain, and eventually Germany, where Hanna is supposed to meet her father at the house of the Brothers Grimm. Meanwhile, Marissa has a CIA team searching for Erik, and she had gotten someone named Isaacs to search for Hanna. (He was a former agent, I guess, and since he's now freelance, he could do things the CIA wouldn't allow Marissa do.) Of course, Marissa herself eventually has first-hand confrontations with both Erik and Hanna, separately.
The reason both Marissa and the CIA were so interested in Erik (a former agent, himself) and Hanna is revealed both by flashbacks and by people talking (somewhat cryptically) about the past. It all has to do with an abandoned CIA project in which Marissa and Erik had been involved, and with Hanna's own true nature, of which she herself was unaware til near the end of the movie. It's definitely an interesting story, though we don't actually get too many details. And it all seems a bit implausible, but that's usually the way with spy movies. Anyway, I don't want to give away anything about the past or how the movie ends, for any of the characters. I'll just say the action scenes were awesome (and I'd love to see Hanna fight Hit Girl from Kick-Ass). But aside from Hanna being a totally amazing badass, it was also quite interesting seeing her learn about real life (there was so much she didn't know, in spite of all Erik's teaching), and about friendship, as well as learning the truth about herself and Erik. There are parts of the film that I could see some viewers finding a bit boring, but personally I was enthralled by the whole thing, from beginning to end.