There are litterboxes that actually can attach to the plumbing in your home and actually flushes about 5 to 10 minutes after your cat has left the box. There is a special litter that's used for it that's harmless to the sewer systems, it's almost like really, really, finely shredded toilet paper pellets.

The cat goes into the litter box, does its business, leaves the box, after a little while you hear the "flush" as it washes itself out with strong jets of water, just like any toilet flushing. The hopper then drops fresh litter into the pan and it's ready for when the cat needs it again.

Unfortunately, these modern self-flushers are NOT reliable, according to people who've tested them. But, in time, I'm sure one that works almost perfectly will be developed.