Here is the full list of all 64 Batman: Arkham City - there are Game of the Year Edition achievements.
Obtain all 72 medals on the original Arkham City maps (as Catwoman)
Obtain all 108 medals on the original Arkham City campaigns (as Catwoman)
Have 12 murderous dates with Calendar Man
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)
Complete every challenge in Arkham City - Main Story, Side Missions, Upgrades, Collectables, New Game Plus and Riddlers Revenge (as Batman)
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)
Complete all 40 of the Catwoman Riddler grid items
Retrieve your stolen goods
Rescue all the hostages from Riddler
Complete New Game Plus
Snap Flash an unarmed thug, an armed thug, an environmental object and a Titan
Destroy all Harley Balloons
Defuse all bombs in 3 minutes or less
Shield Bash 5 different thugs
Use 5 different Quickfire gadgets in one fight as Robin in Harley Quinn's Revenge
Rescue the fifth hostage from Riddler
Stop all assaults in Arkham City
Track down the serial killer
Rescue the fourth hostage from Riddler
Stop the phone booth killer
Reveal the mystery watcher
Stop the contract operative
A moment of remembrance
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)
Perform a perfect combo including all of Batman's combat moves (any play mode)
Obtain 48 medals on the original Arkham City maps (as Batman)
Destroy it all
Collect all of Batman's gadgets and upgrades
Rescue the third hostage from Riddler
Complete all augmented reality training exercises
Reunite the separated couple
Rescue the second hostage from Riddler
Rescue the first hostage from Riddler
We are legion
A deadly game of hide and seek
Fight for survival
Don't look down
All the world is a stage
Time is running out
Solve the first riddle
Obtain 24 medals on the original Arkham City maps (as Batman)
Use 5 different Quickfire gadgets in one fight (any play mode)
Steal the score of a lifetime
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
Uncover the secret of Arkham City
Stop the unstoppable
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
There is only one way in
Save the medical volunteers
Save the damsel, but is she in distress?
Become the Bat