In this talk we shall discuss certain reductions of the discrete KP hierarchy to 1+1 dimensional integrable lattice systems.The connection between these reduced systems and integrable cellular automata (so-called Box and Ball systems) will be commented upon and a general construction will be presented.The systems resulting from this construction provide novel examples of so-called Yang-Baxter maps (i.e., of set-theoretical solutions of the Yang-Baxter equation).If time permits, we shall also discuss the fundamental symmetries of these systems.