I was an adult when the Harry Potter books came out. Around the time of the third or fourth book, my SO introduced me to them. All the hype surrounding the books made me a little skeptical, but once I started reading them, I was surprised at how imaginative they were.
I think it's the detailed characters and settings which make the Harry Potter books special. Most characters, from Harry's friends to the Minister of Magic, feel unique. They're quirky, funny and/or revolting, but they almost always engage you emotionally in some way. Hogwarts is a place which is clever, mysterious, and, above all, fun.
There's an element of wish fulfilment to the books which I think is very appealing to kids. Harry is a mistreated and diminished orphan, who turns out to be better than the family he stays with. Instead of being forced to go to a boring school, he goes to the most exciting school imaginable - ghosts, talking portraits, secret rooms, and so on. Instead of dull classes with droning teachers, he has classes where things explode, come to life or you get to meet mythical creatures. And he gets to do magic! He can make things happen by waving his hands and speaking a few words. My sister-in-law, who went to an afternoon showing of a Harry Potter film, described how the kids in the theatre waved their hands trying to mimic the magic on the screen.
Things like the school classes, the excursions, the teachers, the magic candy, the magic trading cards, and so on, very closely mirror the child's everyday reality, while at the same time making it more magical and exciting.
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