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.Comment: 22 pages. To appear in the volume "Surveys in Geometric Analysis and Relativity celebrating Richard Schoen's 60th birthday.
To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.