Dungeons are where the majority of the action in Trinity Universe takes place. Story dungeons will (obviously) show up when the story requires it (although the game will sometimes give you "free time" before one will show up, requiring you to spend time either resting at the inn or exploring other dungeons), while other dungeons will randomly float in like other objects on a daily basis. Dungeons will have different treasures, managraphics, hidden treasures, and sometimes different enemies depending on what time of day it is when you enter the dungeon. Dungeons also have multiple distinct "floors", each with their own map, and you'll gain access to floors with higher monster levels and better treasures as your Universe Rank increases.
Aside from Yoma Castle, dungeons fall into several distinct categories, with each category playing host to particular "families" of items. For convenience, the dungeon list will be subdivided by these categories.
Yoma King Castle
To learn about the True Route, refer here.
Note: Dungeons marked with * are event dungeons, and do not reappear. Make sure not to miss any IM's or treasures!