A quest is a mission that usually consists in finding one or a series of items. An inner-quest is a journey toward an objective. Characters in quest for their self are often submitted to self-challenging situations.

A quest is usually a collection of tasks that someone requires to complete to reach an objective.
In adventure video games, a quest is a whole sequence of game or a level:
The emphasis of these games is put on the on-screen character's self-development. To go through the game and to improve the qualities of his character, the player must engage it in a series of quests, like finding an item, questioning inhabitants of a village, fighting against bad guys, delivering a princess, gathering for apples...
If the character fulfills a series of quests, he'll be rewarded with experience points, and skills.

In literature, questing for a self-purpose ending or for a material item is often linked.
In children books for instance, the hero is often pictured as a young child who evolves through the story in quest of his identity. He'll win his identity through childish or serious short-term quests, like that of finding a treasure or a lost relative.
In the end the material quest for an item has less value for the character than the journey he took toward his identity.

The most famous quest is that of the Holy Grail. From the 13th century, the spiritual quest for the Grail is at the core of many art works.
The third sequel of Indiana Jones for instance depicted the journey of Indi and his father on the track of the holy cup.