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)
Clear the airwaves
Track down the serial killer
Stop the phone booth killer
Time is running out
Complete all augmented reality training exercises
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
Complete all 40 of the Catwoman Riddler grid items
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)
Obtain 24 medals on the original Arkham City maps (as Batman)
Complete every challenge in Arkham City - Main Story, Side Missions, Upgrades, Collectables, New Game Plus and Riddlers Revenge (as Batman)
Drop in on an old friend
Hammer the point home
There is only one way in
Save the medical volunteers
Become the Bat
Save the damsel, but is she in distress?
No crimefighter should be without this
Last man standing
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
Reveal the mystery watcher
Destroy it all
Reunite the separated couple
Collect all of Batman's gadgets and upgrades
Stop all assaults in Arkham City
Solve the first riddle
Rescue all the hostages from Riddler
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)
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)
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