From Dungeons of Dredmor Wiki

The three basic elements that comprise a quest are:

• Find an item
• Kill a named monster or group of monsters
• Use an item on the Monolith, Mellow Altar or Statue of Eyes

Quests may make use of combinations of these elements, for example, the player may be asked to find an item and then use that item on the Monolith. The items and/or monsters that are the targets of quests do not appear on the map until the quest is accepted, so they may appear in areas already explored. The object of each quest will be located on the same dungeon level as the Statue of Inconsequentia that gave it.

Once a quest is completed, a random named Artifact reward is immediately offered to the player.