These notes represent an updated version of a course on Hamilton's Ricci flow that I gave at the University of Warwick in the spring of 2004. I have aimed to give an introduction to the main ideas of the subject, a large proportion of which are due to Hamilton over the period since he introduced the Ricci flow in 1982. The main difference between these notes and others which are available at the time of writing is that I follow the quite different route which is natural in the light of work of Perelman from 2002. It is now understood how to ‘blow up’ general Ricci flows near their singularities, as one is used to doing in other contexts within geometric analysis. This technique is now central to the subject, and we emphasise it throughout, illustrating it in Chapter 10 by giving a modern proof of Hamilton's theorem that a closed three–dimensional manifold which carries a metric of positive Ricci curvature is a spherical space form.

Aside from the selection of material, there is nothing in these notes which should be considered new. There are quite a few points which have been clarified, and we have given some proofs of well–known facts for which we know of no good reference. The proof we give of Hamilton's theorem does not appear elsewhere in print, but should be clear to experts. The reader will also find some mild reformulations, for example of the curvature pinching results in Chapter 9.