This is a survey of embeddings of graphs in surfaces and their symmetry
properties, with emphasis on those whose automorphism group is transitive
on edges. Such maps arise as regular covers, by their automorphism group A,
of the 14 basic maps with one edge described by Graver and Watkins in the
AMS Memoirs in 1997. The aim is to describe the possible automorphism groups
A for each of the 14 corresponding classes of edge-transitive maps. In the
case of the most symmetric class, consisting of the regular
(flag-transitive) maps, this is related to results of group-theorists such
as Mazurov, Nuzhin and others on generators for finite simple groups. In the
case of the second most symmetric class (the orientably regular chiral maps)
there are links with results of Leemans and Liebeck on automorphism groups
of polytopes. The first half of the talk consists largely of simple examples
from these 14 classes; the second half requires a little group theory
(but not much!). A preprint is available at arXiv:1605.09461v3.