Abstract: Ricci flow is a geometric evolution equation, analogous to the heat equation, that when applied on a Riemannian manifold deforms the metric smoothing out any irregularities of the metric in the process.

Hamilton showed that Ricci flow on 3-manifolds with positive Ricci curvature exists for all time and converges to a constant-curvature metric. This can be generalised to different dimensions and non-positive curvature. In this presentation we will use the techniques derived by Hamilton to talk about the proof of the Uniformization Theorem which states "any closed Riemannian surface has a conformal metric of constant curvature".