From Diablo Wiki
Quests will see a big revamp in Diablo III. Not only is the D3 Team now using all their UI improvement experience from WoW, they are also both streamlining and deepening the story line of the game.
Less quests will be mandatory, and a lot of "opt-in" material and quests will be added.
In the BlizzCon 2008 demo, three quests were available:
Find out more about this old feature with brand new functionalities like the new the new adventure/event features in the Quests article.