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There are a lot of action movies that take themselves "seriously" despite the absurdity of their premises. That's fine, and indeed, a big part of what we like about fiction: their ability to take situations that are just plain unrealistic and make them seem believable.
And this scenario is indeed unbelievable. A 12-year-old girl comes home to find that her family has been murdered. Scared and angry, she knocks on the door of the only person in the apartment complex who was nice to her: the expert assassin Leon. He lets her hide out with him, and she demands he help her get revenge, offering to do anything he needs. He eventually agrees, even going so far as to teach her how to assassinate.
That's some semi-typical unrealistic action movie stuff right there. (The "semi" part being the involvement of a kid in the assassinations - that's more original.) Leon's hits are carried out in a fashion that could only work in action movies - he manages to outwit and take down entire squads that are out to get him, and while it's true that he wins using brains moreso than brawn, it's still rather ridiculous. But it's fun.
What elevates this movie beyond simple mindless trash is two things: its character and emotion.
Leon the hitman's relationship with an orphaned preteen is rather implausible, but it's handled surprisingly well for the most part. Leon starts out being all-business, wanting Mathilda to just get out, and reluctantly agrees to her offer to help him... by basically being his housekeeper, buying his groceries, etc. But he later comes to accept the idea of teaching her to fight, to use a sniper rifle, and later, letting her join him on his assassinations.
With Mathilda's home being no longer safe, she and Leon live together. Mathilda adapts well to this arrangement, but then, she never felt close to her family anyway save for her little brother, the only family member whose death upsets her. She and Leon start to grow closer together as we see scenes of them goofing around, having conversations and making each other laugh.
They begin to bond, although they have very different views of the relationship. Leon sees Mathilda as being more like a daughter or a friend. Mathilda, to Leon's (and the test audience's) horror, sees him as more of a boyfriend, resulting in a few scenes that were removed from the original US version of this French film. (Additional removed scenes are more innocuous: one is Mathilda and Leon carrying out hits together, and another is Mathilda paying off some bullies with stolen money.)
The emotion of the characters is what brings the story to life and keeps it from degenerating into pure trash. On the face of it, a Badass and Child Duo assassinating drug lords and other scum is braindead garbage. But amusingly, toss in enough Character Development and fleshing out, and suddenly, it becomes a movie classic.
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