Act III Bounties
From Diablo Wiki
Most Events show up as Bounties. This is a good way to earn the event achievements because you don't have to search for them.
These Bounties require you to kill monsters on the lowest level of the dungeons listed. They all take the form of Clear "Name of Dungeon"
Kill Bounties require the player kill the named unique monster and some number of other mobs in the same area. After a period of time, an arrow appears in the direction of the unique.
Exploration Bounties require the player to explore a map, defeat 5 groups of enemies and interact with an object they were guarding. Act III has them in the following levels: