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