  The commuting graph of a non-abelian group is a simple graph in which the
vertices are the non-central elements of the group, and two distinct vertices
are adjacent if and only if they commute. In this paper, we classify (up to
isomorphism) all finite non-abelian groups whose commuting graphs are acyclic,
planar or toroidal. We also derive explicit formulas for the genus of the
commuting graphs of some well-known class of finite non-abelian groups, and
show that, every collection of finite non-abelian groups whose commuting graphs
have the same genus is finite.
