abstract: Given a surface or a manifold, the Ricci flow is a well-known technique for deforming its metric, trying to reach a new metric with better properties. Specifically, it is a system of Partial Differential Equations. We will examine it in the simplest case, metrics on a surface, and discuss its main properties from both the analytic and the geometric perspective. This is intended as an introductory course for undergraduates, so the only pre-requisites will be (i) the geometry of curves and surfaces, and (ii) basic complex analysis.