Let G be a nite nonabelian simple group and let be a connected undirected Cayley graph for G. The possible structures for the full automorphism group Aut are specied. Then, for certain nite simple groups G, a sucient condition is given under which G is a normal subgroup of Aut. Finally, as an application of these results, several new half-transitive graphs are constructed. Some of these involve the sporadic simple groups G =J 1 ,J 4, Ly and BM, while others fall into two innite families and involve the Ree simple groups and alternating groups. The two innite families contain examples of half-transitive graphs of arbitrarily large valency.