This game was a surprise success in 2009. I have never expected this game to be extremely good, as most games adapted from superhero media are either mediocre or definitely bad.
Story
Batman and Joker stars in this game, along with Harley Quinn, Killer Croc, Scarecrow, and other supervillains.
Batman has caught Joker, who escaped from Arkham Asylum and escorted him personally to the asylum itself to prevent any escapes from Joker. However, Batman felt that capturing Joker is too easy, and soon found out that it was a plan of Joker's, and he was planning to take over the Arkham Island. It is up to Batman to catch him and stop his evil plans.
Undoubtedly, this game offers the best superhero story ever. Awesome storyline, great characters' development, and plenty of humour by Joker.
I would like to give special attention to every characters' voice actors, as they did extremely good job on carrying their roles in the game, which allows the story to be very awesome and interesting. Joker especially, is one of the highlights of the game that made the story so immersible and great. I would dare say this game has a great storyline that is comparable with The Dark Knight film's storyline, except with less confusions. Without these great voice actors, I highly doubt the game's story will be good as it is now.
Gameplay
Batman can easily fights large number of thugs himself, like his incarnation in Batman Begins and The Dark Knight movies.
The battle gameplay is quite simple. You can punch/kick your enemies using the Strike Button, counter enemy's attack using the Counter Button, stun your enemies using the Stun Button, and dodge your enemy's attacks using the Dodge button. The controls are simple, but the thrill of attacking them is very immersible and addicting. He can also use Gadgets while fighting the enemies, such as the notorious Batarang and Batclaw.
Upgrade Batman's combat kills to take down enemies faster, or upgrade his equipments to destroy optional obstacles for solving Riddler's riddles and collecting collectibles.
This game features Experience (XP) points system to increase Batman's skills and upgrade equipments. By accumulating a certain amount of XP, you can upgrade one Batman's Skill Slot. For example, by getting 1 Upgrade, you can upgrade his combat skills, armor, or his gadgets. Upgrading his combat skills allows him to defeat his enemies faster using special Combo Skills, such as Throw and Instant Takedown. Upgrading his Armor allows him to have more Health. Whereas upgrading his Gadgets such as Batarangs allows Batman to use Batrangs during fights to increase his Combo Counter, etc.
Needless to say, when you have accumulated enough XP to upgrade Batman's abilities, do it so, as it is vital to increase your chances on fighting your enemies, especially in Hard Mode.
If you are able to strike your enemies without missing one and without getting hit, you will initiate a Combo Hit Count, whereas more hits will increase your XP gain. Also, you can only use special moves like Instant Takedown after getting 5 hits on your enemies. It is recommended that you keep building up your Combos by countering your enemy's attacks, striking them efficiently, dodge their attacks, etc to increase XP gain as well as XP bonus. Using a variety of moves in one Combo also allows you to have XP bonus.
The Marker in your Map tells you clearly where you should go. Should you are lost, use the Map to find your way to your next Objective.
The game's exploration is similar to Yakuza 3, or perhaps like other sandbox adventure games such as Assassin's Creed games, but to smaller degree since the map is much smaller compared with those games. You need to go to the Marker's location in your Map to initiate your next Objective. Every Objectives are varied and different, which will not bores you easily. However, there are a few obstacles you must overcome in order to proceed to your next Objective.
Detective Mode allows you to find any alternative route to sneak in buildings, obstacles to overcome, etc. Your Gadgets will be your best friends to solve this kind problems of as well for your Objectives.
Detective Mode is a special feature implemented in this game, where you can check your enemy's condition, strange spots such as cracked walls, etc. You also need to use the Detective Mode to follow someone's trail to the next Objective. It is indispensable as well as your Gadgets.
Your Gadgets are extremely useful to solve obstacles in the game as well as traversing to different areas. For example, Batclaw allows you to reach higher areas, and Explosive Gel allows you to destroy any weak walls as well as take out your enemies. By using your Gadgets effectively, this game is quite easy to beat, even in Hard Mode.
Stealth sections in this game are awesome. I can't help but laugh everytime Joker talks on the PA system.
When encountering enemies, there are 2 types of fighting sections. One, is by beating the crap outta them by using your melee attacks. Two, by taking them out by stealth. When there are enemies use firearms, it is advisable to take them out by stealth, to conserve your health as well as to avoid detection.
Whenever there is a large room filled with armed thugs, it is always recommended to take them out by stealth instead of fighting them head on, as Batman is not invulnerable against bullets. There are few types of stealth takedowns, which are Inverted Takedown, Silent Takedown, and Corner Takedown. These three takedowns are primarily used to takedown your enemies silently without alerting others. Inverted Takedown can only executed when you are on top of a gargoyle while the enemy is directly below of you. However, you need to have the Upgrade for this takedown in order to use it.
As armed enemies are dangerous, you need to hide in well hidden spots, such as vents, gargoyles, etc. You need to plan properly to take out the thugs effectively and efficiently. You can use Glide Kick from tall areas or gargoyles to knock an unsuspected enemy, then use the Ground Takedown to take him out, then quickly escape to a gargoyle using the Batclaw. Or you can use Inverted Takedown to take out an enemy, then lure other enemies to him while you escape in the darkness.
As you progress more into the game, the enemies will be more smarter and more dangerous, so Upgrades are vital to overcome this. For example, some thugs used knives to attack you, whom you cannot attack him directly until you stun him using the Stun button, and etc.
Boss fights in this game are creatively designed in terms of gameplay and levels.
Boss fights are extremely creative designed as well as challenging. I am actually having fun fighting the bosses, as they needs a strategy to defeat them. If you are dead when fighting them, the game always provide the solution on how to defeat them. However, I am slightly disappointed with the last boss fight in the game, as it doesn't provide much challenge like other boss fights.
In short, the gameplay for this game is absolutely awesome. My only complaints are the slightly disappointing last boss fight, as well as unable to fight more supervillains.
Story

Batman and Joker stars in this game, along with Harley Quinn, Killer Croc, Scarecrow, and other supervillains.
Batman has caught Joker, who escaped from Arkham Asylum and escorted him personally to the asylum itself to prevent any escapes from Joker. However, Batman felt that capturing Joker is too easy, and soon found out that it was a plan of Joker's, and he was planning to take over the Arkham Island. It is up to Batman to catch him and stop his evil plans.
Undoubtedly, this game offers the best superhero story ever. Awesome storyline, great characters' development, and plenty of humour by Joker.
I would like to give special attention to every characters' voice actors, as they did extremely good job on carrying their roles in the game, which allows the story to be very awesome and interesting. Joker especially, is one of the highlights of the game that made the story so immersible and great. I would dare say this game has a great storyline that is comparable with The Dark Knight film's storyline, except with less confusions. Without these great voice actors, I highly doubt the game's story will be good as it is now.
Gameplay

Batman can easily fights large number of thugs himself, like his incarnation in Batman Begins and The Dark Knight movies.
The battle gameplay is quite simple. You can punch/kick your enemies using the Strike Button, counter enemy's attack using the Counter Button, stun your enemies using the Stun Button, and dodge your enemy's attacks using the Dodge button. The controls are simple, but the thrill of attacking them is very immersible and addicting. He can also use Gadgets while fighting the enemies, such as the notorious Batarang and Batclaw.

Upgrade Batman's combat kills to take down enemies faster, or upgrade his equipments to destroy optional obstacles for solving Riddler's riddles and collecting collectibles.
This game features Experience (XP) points system to increase Batman's skills and upgrade equipments. By accumulating a certain amount of XP, you can upgrade one Batman's Skill Slot. For example, by getting 1 Upgrade, you can upgrade his combat skills, armor, or his gadgets. Upgrading his combat skills allows him to defeat his enemies faster using special Combo Skills, such as Throw and Instant Takedown. Upgrading his Armor allows him to have more Health. Whereas upgrading his Gadgets such as Batarangs allows Batman to use Batrangs during fights to increase his Combo Counter, etc.
Needless to say, when you have accumulated enough XP to upgrade Batman's abilities, do it so, as it is vital to increase your chances on fighting your enemies, especially in Hard Mode.
If you are able to strike your enemies without missing one and without getting hit, you will initiate a Combo Hit Count, whereas more hits will increase your XP gain. Also, you can only use special moves like Instant Takedown after getting 5 hits on your enemies. It is recommended that you keep building up your Combos by countering your enemy's attacks, striking them efficiently, dodge their attacks, etc to increase XP gain as well as XP bonus. Using a variety of moves in one Combo also allows you to have XP bonus.

The Marker in your Map tells you clearly where you should go. Should you are lost, use the Map to find your way to your next Objective.
The game's exploration is similar to Yakuza 3, or perhaps like other sandbox adventure games such as Assassin's Creed games, but to smaller degree since the map is much smaller compared with those games. You need to go to the Marker's location in your Map to initiate your next Objective. Every Objectives are varied and different, which will not bores you easily. However, there are a few obstacles you must overcome in order to proceed to your next Objective.

Detective Mode allows you to find any alternative route to sneak in buildings, obstacles to overcome, etc. Your Gadgets will be your best friends to solve this kind problems of as well for your Objectives.
Detective Mode is a special feature implemented in this game, where you can check your enemy's condition, strange spots such as cracked walls, etc. You also need to use the Detective Mode to follow someone's trail to the next Objective. It is indispensable as well as your Gadgets.
Your Gadgets are extremely useful to solve obstacles in the game as well as traversing to different areas. For example, Batclaw allows you to reach higher areas, and Explosive Gel allows you to destroy any weak walls as well as take out your enemies. By using your Gadgets effectively, this game is quite easy to beat, even in Hard Mode.

Stealth sections in this game are awesome. I can't help but laugh everytime Joker talks on the PA system.
When encountering enemies, there are 2 types of fighting sections. One, is by beating the crap outta them by using your melee attacks. Two, by taking them out by stealth. When there are enemies use firearms, it is advisable to take them out by stealth, to conserve your health as well as to avoid detection.
Whenever there is a large room filled with armed thugs, it is always recommended to take them out by stealth instead of fighting them head on, as Batman is not invulnerable against bullets. There are few types of stealth takedowns, which are Inverted Takedown, Silent Takedown, and Corner Takedown. These three takedowns are primarily used to takedown your enemies silently without alerting others. Inverted Takedown can only executed when you are on top of a gargoyle while the enemy is directly below of you. However, you need to have the Upgrade for this takedown in order to use it.
As armed enemies are dangerous, you need to hide in well hidden spots, such as vents, gargoyles, etc. You need to plan properly to take out the thugs effectively and efficiently. You can use Glide Kick from tall areas or gargoyles to knock an unsuspected enemy, then use the Ground Takedown to take him out, then quickly escape to a gargoyle using the Batclaw. Or you can use Inverted Takedown to take out an enemy, then lure other enemies to him while you escape in the darkness.
As you progress more into the game, the enemies will be more smarter and more dangerous, so Upgrades are vital to overcome this. For example, some thugs used knives to attack you, whom you cannot attack him directly until you stun him using the Stun button, and etc.

Boss fights in this game are creatively designed in terms of gameplay and levels.
Boss fights are extremely creative designed as well as challenging. I am actually having fun fighting the bosses, as they needs a strategy to defeat them. If you are dead when fighting them, the game always provide the solution on how to defeat them. However, I am slightly disappointed with the last boss fight in the game, as it doesn't provide much challenge like other boss fights.
In short, the gameplay for this game is absolutely awesome. My only complaints are the slightly disappointing last boss fight, as well as unable to fight more supervillains.