Batman: Arkham City is without doubt one of the best games around. Improving on Arkham Asylum, in Arkham City you become Batman and CatWoman, changing roles throughout the game.
In Arkham City, you face the challenge of keeping the criminals in line and finding out the motives behind this new prison complex. Arkham City has a great storyline which allows players to take on some of Batman's most famous villains such as the Joker, Penguin etc... In terms of game design, Arkham City has beautiful graphics with amazing Batmanesque views. Arkham City has a number of challenging but fun puzzles to solve through the games including extra side missions, where you would have to use your skills of deductions and Batman's array of gadgets to solve them. Batman also gets an experience points system which you can use to upgrade stuff like weapons, armour and get new moves. Through the game, you will also get to play as CatWoman. Similar to Batman you get an experience points systems but your goal is different rather than beating up criminals it requires use of Catwoman's dexterity and stealth to achieve the goals.
Overall, I think Arkham City blends gameplay and the storyline to create an amazingly entertaining game which if you include the side missions means hours of addictive fun.