Wow, it's great and I really recommend it.
The fighting mechanics are very well done and it really feels good to punch baddies to the face. Also the fact that many of the gadgets are usable in many types of situations (ie. Fights, Puzzle solving) creates a nice feel of variety to the game.
This game reminded me a bit of Sly 3 on PS2, because you can fly above a city that you are free to explore. The world is Just a lot bigger in Arkham City. Also more mature and dark. I really like the atmosphere there.
Then there's the characters. Oh man, they're so well created. I specially liked the Joker and Harley Quinn. I'm not a big Batman fan, I have seen at least one Batman movie, but I don't really remember what any of the characters sounded like in them. Still I really, really liked their voices. In fact I liked them so much, I would play the game again just for that.
Speaking about playing the game again, there's this great thing in this game called "New game plus" which let's you start the game all over again, with tougher enemies. To keep the balance though, you can keep all your gadgets and upgrades from the first playthrough.
Overall, I think this was a great buy and you'll probably like it if you liked Sly (Especially Sly 3) when you were younger, or just generally enjoy the action/adventure genre.