I was have a nosey at the Museum of Flight's website and it occured to me that one could make a solar powered UAV and send it off to circumnavigate the world.
With enough money, yes, this could be done - the better question (other than to just prove it can be done) - is why? What greater purpose could it serve?
It'd be a bit tricky, you'd have to design the frame carefully to provide lots of lift for minimum thrust, you'd have to balance the power so it's a low load which can run off battery for quite a while (if you sent it over the poles it'd have to cope with periods of more dark than light), that kind of thing.
You'd probably need something big, similar to the Gossamer Condor/Albatross wing area, to have enough area for the solar cells; plus you'd need to find lightweight solar cells. It could probably be done, but it wouldn't be cheap or easy.
If one's goal were just to circumnavigate the world autonomously, then trying to speak with the Rutan's about their Voyager project might open some doors (if you had a good design and some money up front, I'd imagine). Not sure nowadays, but back before Voyager flew it was fairly easy to just walk into the hanger in Mojave...
Then it struck me that a boat would be simpler. You wouldn't have to worry about the weight as much, you'd need fewer instruments (no tilt, altitude or airspeed sensors) and if it ran out of power it could just sit and wait til it was sunny again, then set off.
Besides the thoughts already given, I think a boat/ship/sub is going to be just as difficult to pull off - as much engineering and monetary sacrifice.
IIRC, drug smugglers have used underwater autonomous subs in bids to ship drugs; I also know that there have been long-range underwater "glider" devices tested (they move forward by sinking and rising), but I don't know if any have done more than tests, or if they are in general use by researchers. I don't think they are solar powered, though (but then again, they don't use a lot of energy). They are suppose to rise to the surface and use satellite communications (likely Iridium or similar) to send/receive data.
There is definitely a market out there for either system...