Robot lawn mowers are very common in Europe. You'll see these small electric mowers in people's yards all over the place. Businesses also have them running all day out front. Never seen one in the US.
They are equipped with GPS, so they are locked to a specific area to prevent theft.
GPS are very fancy ones, unless that's changed since I last looked into it. A buried wire, "invisible fence", has been the norm for all consumer grade ones I've seen.
I think the wire ones usually do too. When they're out of power and possibly if it rains, they go straight until they reach the wire, and then follow it home to dock.
Yes, I believe the GPS ones are also guided by a wire to find edges and find their way home. The GPS is mostly for theft prevention. Won't work outside a specific area unless it's unlocked from the backend.