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