As usual, I'll start with the plot. Some months of the first game, Quincy Sharp, the warden of Arkham Asylum in, well, Arkham Asylum, is now the mayor of Gotham City, and has resolved to clean up crime in the best way possible: section off a large chunk of the city and turn it into Arkham City, a mega-prison in which the prisoners are free to roam around, fighting and killing one another. Sharp appoints reccuring weirdo Hugo Strange as warden of the facility. At the start of the game, Bruce Wayne, Batman's alter-ego, suspects a conspiracy and gets himself arrested by going to the prison and speaking out against it. Presumably, he also kills several thousand people while there, because he is immediately thrown into the prison without so much as a trial. That, or the legal system in Gotham City makes about as much sense as a neon-filled whale, as Strange is apparently to throw people in prison just because they annoy him.
Despite the questionable introduction, this does allow for an excellent first sequence in which the player, as Wayne, must fight through a corridor of bloodthirsty inmates before encountering The Penguin, now a cockney for some reason, and punching his face in, followed by an escape to a nearby roof to retreive Wayne's batgear. This part of the game is highly cinematic and sets up the rest of the game well.
Gameplay in Batman: Arkham City is solid, and mostly revolves around three elements: exploration, combat and stealth. Combat is simple and works incredibly well. Press X to punch, kick and generally assault the nearest enemy, while Y counters, B stuns enemies and A allows you to vault over enemies. The four buttons can be combined to create some satisfying combos as Batman leaps around a room, knocking out enemies and hearing their bones loudly being cunched into dust. Batman, as in the comics and movies, has a number of gadgets, many of which are unlocked from the start, some unlocked later on via an RPG-style XP system. They can be used in combat, or used for exploration, such as the Line Launcher, which allows Batman to traverse gaps in terrain.
They can also be used in the fantastic stealth parts of the game, during which Batman must silently makes his way around an enclosed area, silently taking out enemies one by one. The high-quality enemy AI and numerous ways of approaching these challenges, such as hanging from a gargoyle and grabbing enemies from above, taking out enemies from within grates in the floor, and even punching through walls to knock foes out make these sections great fun.
Variations of both the combat and stealth sections can be played in the additional Challenge mode, in which the player can either take out enemies in combat while avoiding taking damage for the most points possible, or eliminate henchmen in a stealth section in the shortest time possible. Buy the game new rather than pre-owned and get the ability to play as Catwoman, who plays similarly to Batman, with some different weapons and the ability to hang from ceilings. New challenges can be unlocked in Story mode, including a cool 2.5D combat challenge set on a moving train. Purchasers of the Robin Edition also get to play as Robin, who, again, plays similarly to Batman, with different weapons. To be honest, the Robin Edition, in my opinion, is somewhat of a rip-off.
One of the major additions to Batman: Arkham City is the fact that the game map is now open-world, set in a sunstantial area, which, while nowhere near the level of Just Cause 2, is expansive and offers plenty of scope for exploration. There are a number of side-missions, such s tracking down a serial killer and rescuing hostages from The Riddler, as well as approximately 400 Riddler challenges, around double the number found in Arkham Asylum. These challenges involve finding things hidden around the game world, though this often requires the completion of a puzzle, preventing this element of the game from becoming just a boring scavenger hunt. Combat is also intergrated into these challenges, as they can be found on the map by interrogating The Riddler's henchmen. As a side note, these interrogations are often hilarious, as Batman promises not to harm the criminal as long they gives up information, at which point they invariably do, and Batman punches them in the face, fracturing their skulls into pieces.
The problem with Arkham City being open-world, though is that it's less focused. In Arkham Asylum there was always a sense of urgency about getting to the next objective, and the best parts of the game were the sections with Scarecrow in them, largely as they could occur at any time without warning. There is nothing like this in Arkham City; when it's up to the player to move the story along, this effect disappears, and the villains pose far less of a threat. Much of the tension of Arkham Asylum is gone.
Boss fights have been siginificantly improved since Arkham Asylum, especially the fight set in a surreal, somewhat Oddworld-like desert. They are quite easy, though, and can almost all be completed in one try without too much difficulty. The final boss fight seems a bit incongruous, featuring a villain who doesn't appear until right at the very end, who can be taken out incredibly easily considering it's the final boss fight. I personally only died once during this fight, and the hint which appeared as a result gave away how to win the fight. After that, it was a walk in the proverbial park.
There are a huge number of villains in Arkham City, some expected, some less so. The Penguin, The Joker and Two-Face are at war over turf in the prison, and this leads to some quite interesting plot points. However, it too removes some of the threat posed by the villains of the game, as they are no longer united against Batman as they were in Arkham Asylum. The vast number of villains also means that some get hardly more screen time than Scarecro in The Dark Knight. Some of these are quite disproportionate - fan favourite Two-Face only appears twice in the whole game for around 2 to 5 minutes at a time, calls Catwoman a bitch, gets effortlessly dispatched and then disappears. Meanwhile, The Penguin is around for about an hour, maybe more, despite arguably being the least interesting main villain in the game. This also applies to other characters, such as Robin, who literally only appears in the Story mode for around a minute before leaving. This also goes for the character of Vicki Vale, who shows up for one brief mssion and then is nowhere to be seen afterwards.
Batman: Arkham City is not entirely without faults, and in fact, is arguably not as good as its predecessor. However, it's still a great game which I highly recommend. The combat and stealth sections are even better than they were in Arkham Asylum, the graphics are great and the story is pretty good. If you only play the story missions, it should provide about 10 hours, with dozens more afterwards for everyone else. Arkham City is a great game for fans of the Batman franchise and action games alike.