There are plenty of Chess AI's around, and evidently some are good enough to beat some of the world's greatest players.
I've heard that many attempts have been made to write successful AI's for the board game Go, but so far nothing has been conceived beyond average amateur level.
Could it be that the task of mathematically calculating the optimal move at any given time in Go is an NP-complete problem?
Chess and Go are both EXPTIME complete. IIRC, Go has more possible moves, so I think it a higher multiple of that complexity class than chess. Wikipedia has a good article on the complexity of Go.