Genus spectra for symmetric embeddings of graphs on surfaces

This paper gives an account of some very recent work by the author in determining all regular maps on surfaces of Euler characteristic −1 to −200 (orientable and non-orientable), observing patterns in the resulting data, and joint work with Jozef Siráň and Tom Tucker in proving the existence of infinitely many gaps in the genus spectrum of regular but chiral maps (on orientable surfaces) and the genus spectrum of reflexible regular maps on orientable surfaces with simple underlying graph.