Let $G$ be a finite nonabelian simple group and let $\Gamma$ be a connected undirected Cayley graph for $G$. The possible structures for the full automorphism group ${\rm Aut}\Gamma$ are specified. Then, for certain finite simple groups $G$, a sufficient condition is given under which $G$ is a normal subgroup of${\rm Aut}\Gamma$. 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 infinite families and involve the Ree simple groups and alternating groups. The two infinite families contain examples of half-transitive graphs of arbitrarily large valency.