Author’s abstract: A Cayley map is an index one voltage graph imbedding, necessarily of a Cayley graph into a closed orientable 2-manifold. A surprisingly large number of the significant constructions of topological graph theory have been Cayley maps. We survey these. The following new result is presented: The complete graph \(K_n\) has a maximum genus Cayley map if and only if \(n\equiv 0\) or \(3\pmod 4\).