The correct playing order for the Batman Arkham trilogy is: "Batman: Arkham Asylum", "Batman: Arkham City", and "Batman: Arkham Knight". Let's take a detailed look at each game in the trilogy:

1. "Batman: Arkham Asylum":

Released in 2009, this game is the first installment in the Arkham series. The story follows Batman as he is trapped inside Arkham Asylum, the psychiatric hospital where the Joker orchestrates a plan to take over Gotham City. Players control Batman and navigate through the asylum, fighting criminals and using detective skills to solve puzzles. The game received critical acclaim for its engaging storyline, innovative gameplay mechanics, and immersive atmosphere.

2. "Batman: Arkham City":

Released in 2011, this game is the second installment in the trilogy. The story takes place one year after the events of "Batman: Arkham Asylum". Gotham City has been transformed into a walled-off prison where criminals roam freely. Batman must navigate through the city, encountering iconic villains such as the Joker, Two-Face, and Mr. Freeze. The game introduces an open-world environment, allowing players to freely explore Gotham City and complete various side missions. It received praise for its expanded gameplay mechanics, improved combat system, and gripping narrative.

3. "Batman: Arkham Knight":

Released in 2015, this game is the final installment in the trilogy. The story follows Batman as he faces his ultimate challenge, the Scarecrow, who threatens to unleash a deadly toxin on Gotham City. As Batman, players must fight against various villains, including the iconic Arkham Knight, while also dealing with personal conflicts and challenges. The game introduces the Batmobile as a playable vehicle, allowing players to navigate the streets of Gotham City in a new exciting way. "Batman: Arkham Knight" received positive reviews for its stunning graphics, immersive gameplay, and satisfying conclusion to the trilogy.

In summary, the Batman Arkham trilogy provides an immersive and thrilling experience for players, allowing them to step into the shoes of the Dark Knight and fight against Gotham City's most notorious criminals. By following the correct playing order of "Batman: Arkham Asylum", "Batman: Arkham City", and "Batman: Arkham Knight", players can fully appreciate the evolving storyline and gameplay mechanics throughout the trilogy.