Here is the full list of all 64 Batman: Arkham City - there are Game of the Year Edition achievements.
Obtain 24 medals on the original Arkham City maps (as Batman)
Reveal the mystery watcher
Stop the phone booth killer
Collect all of Batman's gadgets and upgrades
Rescue the third hostage from Riddler
Rescue all the hostages from Riddler
Obtain 48 medals on the original Arkham City maps (as Batman)
Obtain all 72 medals on the original Arkham City maps (as Batman)
Obtain 78 medals on the original Arkham City and Robin Bundle Pack maps (as Robin)
Obtain 78 medals on the original Arkham City and Nightwing Bundle Pack maps (as Nightwing)
Rescue the fourth hostage from Riddler
Rescue the fifth hostage from Riddler
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
Have 12 murderous dates with Calendar Man
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
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 114 medals on the original Arkham City and Robin Bundle Pack campaigns (as Robin)
Obtain 114 medals on the original Arkham City and Nightwing Bundle Pack campaigns (as Nightwing)
Track down the serial killer
Stop the unstoppable
A moment of remembrance
Use 5 different Quickfire gadgets in one fight (any play mode)
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
Destroy all Harley Balloons
Become the Bat
Save the damsel, but is she in distress?
Save the medical volunteers
There is only one way in
Hammer the point home
Clear the airwaves
Last man standing
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
Destroy it all
Time is running out
Reunite the separated couple
Complete all augmented reality training exercises
Stop all assaults in Arkham City
Solve the first riddle
Rescue the first hostage from Riddler
Rescue the second hostage from Riddler
Complete one combat challenge without taking damage (any character)
Perform a perfect combo including all of Batman's combat moves (any play mode)
Drop in on an old friend
Steal the score of a lifetime
Retrieve your stolen goods
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
Use 5 different Quickfire gadgets in one fight as Robin in Harley Quinn's Revenge