Hunts are the main optional sidequest in Final Fantasy XII. They entail players hunting down stronger monsters, which are usually terrorizing a person or town. There are two ways you can check on Hunts (called Marks or Elite Marks), which are notice boards in specific towns, or by talking to Montblanc in the Clan Centurio guild in Rabanastre.
When you first find out about a Mark from Montblanc or the notice board, you must hunt down the petitioner and speak to him/her before you can go out to actually track down the Mark. You aren't given many clues about a Mark's location, as you are given hints by the petitioner. Once you hunt down the Mark and vanquish it, go back to the petitioner and you'll receive your reward. There are a total of 32 normal Marks and 13 Elite Marks, making a total of 45 Marks in the game.
