We've been paper free for 3 or 4 years now. For my own family I don't wrap at all. I take all their gifts out of any packaging, make sure everything that needs assembling is put together and set it all up under the tree. I do hang gift tags off of the things meant for just one child as opposed to shared gifts. The kids aren't any less excited Christmas morning. It's just as much fun to take in everything under the tree and sort through it all. They don't even notice the lack of wrappings. I love it because there is no clean-up to do.
For my parents, we've been regifting the same gift bags (both cloth and paper) back and forth for about 10 years now.