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