Hyperbolic tilings of type {3,7}, {7,3}, {4,5}, and {5,4} are mapped onto closed 2-manifolds of genus 3 through 7 with as much symmetry as possible. All these maps exhibit local regularity which makes all vertices look equivalent, but most lack some of the global flag-transitive symmetries that are topologically possible. On the other hand, many of these mappings allow some nice symmetrical embeddings in 3D Euclidean space with tiles of limited deformation, which makes it possible to create attractive Escher-like tiling patterns on these surfaces of higher genus.