Can you paint over wall paper instead of removing it first?

Yes you can as long as it is cleaned with a soap and water mixture and primmed before applying the actual paint.

Make sure that all the wallpaper is securely adhered as well. The moisture in the paint sometimes lifts some of the wallpaper. As rankrank55 says... preparation is ultimate.

You Can But What Will Happen Is The Moisture In The Paint Will Losing Some , Not All, But A Few Places On The Wall. In Essence You Have Some Bubbles That Will Come Up, You'll Have To Cut Them Out With A Razor Knife And Then Mud Over Them A Few Times And Then Sand, Prime And Paint Over The Spot. You Will Also See All The Seams Where The Wall Paper Meets The Other.
If You Don't Mind This, Ok But If You Do You'll Need To Treat Each Joint As A Sheetrock Joint, With Tape And 3 Coats Of Mud. Sand ,prime And Paint

You will be able to see the seams unless you do something about that before painting. We found it better in the long run to remove the paper, then paint.

How can the wall paper be removed? Or should it be left in place?

If the paper wasn't strippable, we peeled off as much as we could and then soaked sections with warm water and peeled those off. We scraped the walls that were under the paper and smoothed them out, spackled to fill holes, smoothed on goop, then primed when that was dry. Then we sanded. And sanded. And sanded.

Hey, gang... I have painted over a lot of my wallpaper in this "lil ol house". I sandpaper the seams, I am doing this right now. If the wallpaper is really well bonded, I haven't had a problem. Do some test areas. Right now, there is a lot of companies that have paintable wallpaper. It has an inherent raised design. Great for rough walls. It's already primed. Install & paint. I am doing a large wall (27x12) right now with faux painting... it looks "Martha Stewart" O.K. Have fun, nothing ventured, nothing gained.

Have to be careful no to use too much water if it is sheetrock underneath. You can damage the paper layer to the point that it is a BIG mess.

I prefer to rent a steamer if it is on sheetrock - you can control how wet it gets a lot easier, only apply enough to loosen and strip.