Given a group G with generators , it is well-known that the set of color-preserving automorphisms of the Cayley color digraph is isomorphic to G. Many people have studied the question of finding graphs (including Cayley graphs) with a given automorphism group G, the graphical regular representation problem. This talk asks a different question: how much larger than G can the full (digraph) automorphism group of a given Cayley graph for G be? The question doesnt have a complete answer yet, so we will survey results known so far. All of these concepts will be defined, and many lovely (and colorful) pictures will be shown.