Here is the full list of all 64 Batman: Arkham City - there are Game of the Year Edition achievements.
Complete every challenge in Arkham City - Main Story, Side Missions, Upgrades, Collectables, New Game Plus and Riddlers Revenge (as Batman)
Complete New Game Plus
Perform a perfect combo including all of Batman's combat moves (any play mode)
Obtain 114 medals on the original Arkham City and Nightwing Bundle Pack campaigns (as Nightwing)
Obtain 78 medals on the original Arkham City and Nightwing Bundle Pack maps (as Nightwing)
Obtain 114 medals on the original Arkham City and Robin Bundle Pack campaigns (as Robin)
Obtain 78 medals on the original Arkham City and Robin Bundle Pack maps (as Robin)
Obtain all 108 medals on the original Arkham City campaigns (as Catwoman)
Obtain all 72 medals on the original Arkham City maps (as Catwoman)
Obtain all 108 medals on the original Arkham City campaigns (as Batman)
Obtain 72 medals on the original Arkham City campaigns (as Batman)
Obtain 36 medals on the original Arkham City campaigns (as Batman)
Obtain all 72 medals on the original Arkham City maps (as Batman)
Obtain 48 medals on the original Arkham City maps (as Batman)
Obtain 24 medals on the original Arkham City maps (as Batman)
Complete one combat challenge without taking damage (any character)
Snap Flash an unarmed thug, an armed thug, an environmental object and a Titan
Have 12 murderous dates with Calendar Man
Defuse all bombs in 3 minutes or less
Destroy all Harley Balloons
Use 5 different Quickfire gadgets in one fight as Robin in Harley Quinn's Revenge
The joke's on who?
Clean up the Dry Docks
Find a way into the secret base
No crimefighter should be without this
Shield Bash 5 different thugs
Zip Kick 5 different thugs
Rescue all the hostages from Riddler
Complete all 40 of the Catwoman Riddler grid items
Rescue the fifth hostage from Riddler
Rescue the fourth hostage from Riddler
Stop all assaults in Arkham City
Retrieve your stolen goods
Collect all of Batman's gadgets and upgrades
Rescue the third hostage from Riddler
A moment of remembrance
Stop the phone booth killer
Use 5 different Quickfire gadgets in one fight (any play mode)
Reveal the mystery watcher
Track down the serial killer
Rescue the second hostage from Riddler
Stop the contract operative
All the world is a stage
Don't look down
Steal the score of a lifetime
Fight for survival
Destroy it all
Complete all augmented reality training exercises
Reunite the separated couple
Time is running out
A deadly game of hide and seek
We are legion
Uncover the secret of Arkham City
Stop the unstoppable
Last man standing
Clear the airwaves
Rescue the first hostage from Riddler
Solve the first riddle
Drop in on an old friend
Hammer the point home
There is only one way in
Save the medical volunteers
Save the damsel, but is she in distress?
Become the Bat