Mobius regular maps are surface embeddings of graphs with doubled edges such that (i) the automorphism group of the embedding acts regularly on flags and (ii) each doubled edge is a centre of a Mobius band on the surface. In the first part of the paper we give an abstract characterisation of Mobius regular maps with a given automorphism group in terms of two dihedral subgroups intersecting in a special way. As an application we exhibit an interesting correspondence between Mobius regular maps of valence 6 and 3-arc-transitive cubic graphs. The second part of the paper deals with constructions of Mobius regular maps on certain classes of simple groups. The main result here is an exact enumeration of all such maps on PSL(2, q) groups. A number of other results related to the two main topics are presented. (c) 2006 Published by Elsevier Inc.