The parabolic analogue of Liouville’s theorem, due to Appell and Hirschman, states that every positive solution to the heat equation on $\mathbb{R}^n\times (-\infty,\infty)$ with subexponential spatial growth is constant. Motivated in part by Appell’s theorem, many recent attempts have been made to classify ancient solutions to nonlinear parabolic equations arising in geometry, such as the heat equation on manifolds, the mean curvature flow of hypersurfaces, and the Ricci flow of Riemannian metrics. We describe some of this progress, some of our own contributions, and some of the outstanding problems.