From the minute 10:05 of video does not have audio. In 1997 Graver and Watkins partitioned edge-transitive maps into 14 classes, distinguished by the quotient of each map by its full automorphism group. These classes include those of regular, chiral and just-edge-transitive maps. I shall describe the possible topological and combinatorial properties of the maps in these classes, including maps with boundary, together with the possibilities for their automorphism groups. In the case of finite simple groups, the latter builds on results of Nuzhin and others relevant to regular maps (with a small but important correction), and more recently of Leemans and Liebeck for chiral maps.