The Yamabe flow is a geometric evolution equation whose goal is to improve the geometry of a manifold by deforming its metric toward one with constant scalar curvature. In this talk, I will introduce the basic background and motivation for the Yamabe flow and then discuss recent developments concerning ancient solutions, namely, solutions that exist for all negative times. In the latter part of the talk, I will present joint work with Dr. Haixia Chen (Hanyang University) and Prof. Monica Musso (University of Bath, UK) on the existence of nonradial ancient solutions.