  Ricci flow on two dimensional surfaces is far simpler than in the higher
dimensional cases. This presents an opportunity to obtain much more detailed
and comprehensive results. We review the basic facts about this flow, including
the original results by Hamilton and Chow concerning Ricci flow on compact
surfaces. The rationale for this paper, however, is especially to survey recent
work concerning this flow on open surfaces, including various classes of both
complete and incomplete surfaces, where a number of striking new phenomena have
been observed.
