H.P.
I have a CleverCat box - it's like a tall rubber tub (kind of like ones you'd store stuff in your garage in, but a little smaller) with a flat lid that has a hole in it. So they get up on top and climb down into it. I love it - we have very little mess around the litter box these days. One complaint - I purchased the liners, but my cats tend to scratch apart the bag when they cover their messes, so I should replace the bag well before I'm ready to replace the litter. Maybe if I cleaned it more often so they didn't have to scrape so much to cover the mess I'd have more success...
I've tried all types of litter box and I'll never buy another kind again!