The first game in the series, Arkham Asylum, was very well received by fans and critics, but one negative of the game was that it mainly took place inside Arkham Asylum and you navigated enclosed areas. In Arkham City, the game is a bit more open including more side-quests to give you that freedom of exploration. In a lot of ways, it is similar to Assassin's Creed Syndicate; there's plenty of side-quests and distractions, which will involve plenty of fighting and sneaking around. To navigate you will be gliding and hook-shotting around the city which is very satisfying.
There are plenty of villains in the game, many returning from the first game, and many more making small cameos. You could say there's a little too many since their stories and motivations aren't well fleshed out; it's mainly fan service.
One year has passed since the events of the first game where The Joker took control of Arkham Asylum. Now the former warden Quincy Sharp has bordered off Arkham City and populated it with inmates, leaving Hugo Strange in charge. Gangs loyal to The Joker, Two-Face and Penguin now fight for control within the city. After Bruce Wayne protests about this, he is captured and thrown into the city. The Joker is ill from the effects of the Titan formula from the events of the first game. After capturing Batman and infecting him, Batman also needs to find a cure.
All the gameplay ideas from the first game are here, so you will be required to use stealth in certain sections, fight against a horde of enemies, or use platforming to traverse the environment. There seems to be even more Riddler trophies scattered around, usually locked behind simple puzzles.
Batman starts off with the majority of the gadgets he had in the first game, but many more are included this time and a lot of them can be used in combat to vary up your attacks. The amount of gadgets you have can be quite cumbersome. I found it hard to remember which button uses which gadget during combat. Outside combat, you are pressing the D-Pad multiple times to cycle through gadgets.
The main combat mechanic remains unchanged, so Batman can dodge, punch, counter and stun enemies. You are rewarded with more XP when you achieve combos, so you are always looking to counter and dodge to avoid the hits. Although Batman can take a good few punches, he doesn't last long against bullets which means you must dispatch gunmen with stealth, or take them out as a priority in combat. As you progress through the game, you are introduced to new enemies such as armoured (which need to be stunned, then attacked with the 'beat-down' move), shielded enemies (where you need to leap over them, then slam into them), and enemies equipped with stun batons (cannot attack from the front).
The Catwoman DLC was bundled into new copies, and certain versions of the game. This allows you to play as Catwoman across four additional story missions which are interspersed through the main story. She has her own set of Riddler trophies, so if you find them with Batman, then you can only tag them to add them to your map. It's easy to control Catwoman as she essentially has an equivalent move-set. She is a more agile version of Batman, and can use her whip to swing around Gotham Spiderman-style. She can crawl on certain parts of the ceiling, and scramble up walls. I enjoyed Catwoman's parts and would have liked even more missions.
Once the credits roll, there is Catwoman's final mission, then you are left to play on (with either character) to find the rest of the trophies and complete remaining side-quests. You can also play a New Game Plus, taking across your upgrades and playing with increased difficulty.
I think Arkham City is a great sequel because it takes everything that made the first game good and builds upon it. The flaws with the game are minor, and it's enjoyable for both hardcore and casual Batman fans.