The study of diferentiable manifolds is a deep an extensive area of mathematics. A technique such as the study of the Ricci flow turns out to be a very useful tool in this regard. This flow is an evolution of a Riemannian metric driven by a parabolic type of partial differential equation. It has attracted great interest recently due to its important achievements in geometry such as Perelman\u27s proof of the geometrization conjecture and Brendle-Schoen\u27s proof of the differentiable sphere theorem. It is the purpose here to give a comprehensive introduction to the Ricci flow on manifolds of dimension two which can be done in a reasonable fashion when the Euler characteristic is negative or zero. A brief introduction will be given to the case in which the Euler characteristic is positive
