Main story line quests and changing classes

If I decide to level up another class past my current level, will the main story line quests apply to my new class?

Basically are the main story line quests character based and you can complete them as any class you happen to level to?

Thanks!

Yes, but you will have to level your current class to the same level before you can continue again. If not it will be red and you can't accept it.

Thanks DharmaXII!