Once you've done a quest in an act you can't do it again, but you can (and often must) do each quest on each difficulty level, getting the full reward each time.
==Quest Eligibility==
To get a quest, you must create a game yourself, or join one created by a character who hasn't done that quest yet. When you first join a game you can open up your quest menu, click to the act you want to check, and see if that quest is active by clicking on it. It will tell you what to do for that quest, or if you can't do it that game because someone else already has or if the game was created by a character who has already done it.
Some quests you can do anyway, even if they say you can't. These include all of the ones where you must kill something or open a chest to obtain some quest item. If you need the item the monster or chest will drop it, even if the quest isn't available that game. You'll get the item, you just won't get credit for the quest, but that doesn't matter in these cases.</font>
=Quest Rewards=