  A vertex-transitive map $X$ is a map on a closed surface on which the
automorphism group ${\rm Aut}(X)$ acts transitively on the set of vertices. If
the face-cycles at all the vertices in a map are of same type then the map is
said to be a semi-equivelar map. Clearly, a vertex-transitive map is
semi-equivelar. Converse of this is not true in general. We show that there are
eleven types of semi-equivelar maps on the torus. Three of these are equivelar
maps. It is known that two of the three types of equivelar maps on the torus
are always vertex-transitive. We show that this is true for the remaining one
type of equivelar map and one other type of semi-equivelar maps, namely, if $X$
is a semi-equivelar map of type $[6^3]$ or $[3^3, 4^2]$ then $X$ is
vertex-transitive. We also show, by presenting examples, that this result is
not true for the remaining seven types of semi-equivelar maps. There are ten
types of semi-equivelar maps on the Klein bottle. We present examples in each
of the ten types which are not vertex-transitive.
