There are 51 Trophies that can be collected.

Bronze

I'm Batman    
Become the Bat     

Acid Bath    
Save the damsel, but is she in distress?     

Savior    
Save the medical volunteers     

Chimney Sweep    
There is only one way in     

One-Armed Bandit    
Hammer the point home     

Communication Breakdown    
Clear the airwaves     

Gladiator    
Last man standing

Lost And Found    
Uncover the secret of Arkham City

Freefall    
Don't look down

Forensic Expert    
Collect enough evidence to locate the gun for hire

Mystery Stalker    
Reveal the mystery watcher

Distress Flare    
Answer the call for help

Ring Ring    
Answer a ringing phone

Stop the Clock    
Time is running out     

Bargaining Chip    
Reunite the separated couple

Fully Loaded    
Collect all of Batman's gadgets and upgrades     

Aggravated Assault    
Stop all assaults in Arkham City     

IQ Test    
Solve the first riddle     

Conundrum    
Rescue the first hostage from Riddler     

Mastermind    
Rescue the second hostage from Riddler     

Puzzler    
Rescue the third hostage from Riddler     

Intellectual    
Rescue the fourth hostage from Riddler

Bronze Revenge    
Obtain 24 medals on the original Arkham City ranked maps (as Batman)     

Silver Revenge    
Obtain 48 medals on the original Arkham City ranked maps (as Batman)

Campaign Bronze    
Obtain 36 medals on the original Arkham City campaigns (as Batman)     

Campaign Silver    
Obtain 72 medals on the original Arkham City campaigns (as Batman)

Flawless Freeflow Fighter 2.0    
Complete one combat challenge without taking damage (any character)

Gotham Base Jumper    
Jump off the tallest building in Arkham City and glide for 1 minute without touching the ground.     

Pay Your Respects    
A moment of remembrance     

Storyteller    
Have 12 murderous dates with Calendar Man     

Catch    
Find someone to play remote Batarang catch with     

50x Combo    
Complete a combo of 50 moves (any play mode, any character)     

Perfect Freeflow 2.0    
Perform a perfect combo including all of Batman's combat moves (any play mode)     

Gadget Attack    
Use 5 different Quickfire gadgets in one fight (any play mode)

Silver

Wrecking Ball    
Stop the unstoppable

Sandstorm    
We are legion     

Hide And Seek    
A deadly game of hide and seek     

Ghost Train    
Fight for survival

Exit Stage Right    
All the world is a stage

Contract Terminated    
Stop the contract operative     

Serial Killer    
Track down the serial killer

Broken Toys    
Destroy it all

Dial Z For Murder    
Stop the phone booth killer

AR Knight    
Complete all augmented reality training exercises

Brainteaser    
Rescue the fifth hostage from Riddler     

Genius    
Rescue all the hostages from Riddler

Gold Revenge    
Obtain all 72 medals on the original Arkham City ranked maps (as Batman)

Campaign Gold    
Obtain all 108 medals on the original Arkham City campaigns (as Batman)

Twice Nightly    
Complete New Game Plus

Gold

Perfect Knight – Day 2    
Complete every challenge in Arkham City – Main Story, Side Missions, Upgrades, Collectables, New Game Plus and Riddlers Revenge (as Batman)

Platinum

Platinum    
Congratulations!

http://www.ps3trophies.org/game/batman-arkham-city/trophies/