A geometric flow is a way of evolving a geometric object (e.g. a smooth closed curve in the plane) in order to improve it in some sense (e.g. to regularise it and make it rounder). Led by Ricci flow, this field has proved to be extremely successful in solving hard problems in other areas, and a typical colloquium might survey some of these applications. In this talk I would like to focus instead on some of the nonlinear PDE phenomena that occur along the way. I plan to illustrate them by focussing on the most visual and non-technical examples.