Final Fantasy XII
About Rare Monsters
When gameplay reaches to Port Balfonheim, return to The Phon Coast. At the Gate Crystal section, talk to a Bangaa captain (Huntmaster) in front the cabin, he's around the South-West section. Now you're starting the first rare monster hunt. Go to the next zone (Phon Coast/The Vaddu Strand) and go all the way east up to the end of the hill, look down at the beach. After awhile, the first rare monster will appear down there near the beach, go down and defeat it, then back to the Huntmaster. Starting now, you can search for more rare monsters in the world of Ivalice. Some might actually show up before you start the quest, but the rare monsters which drop the Engage/Trophy Items will only spawn after starting the quest.
Rare monsters which drop Engage items (name Trophy for US version) would only appear once, after defeating them, they will never appear again. Collect Engage items and you can trade them for other things from the Bangaa race in Phon Coast. The Hunt Shop nea there will sell different items depend on which Bangaa has the Engage items.
About the items dropped from the monster, a few of them you can steal the items which are properly drop after you have kiiled it, but if you do so, it won't drops the stolen item after defeating it.
Stealing items from rare monsters is much more harder than other enemies. Item drops from rare monsters are rare and random, except Engage/Trophy Item, sometimes they might drop nothing after defeating them. But keep on trying, you could get some rare or useful items from them.
Most of the rare monsters can also drop Arcana and High Arcana (check the monographs section of the site for more on this), I guess don't need to write them down. And the monster's HP might not listed correctly since it's counted by myself and the monsters can appear with different level.
Now that you know what they are, check out the Rare Monsters Guide here.