What is a canonical geometry? To answer this question, one first needs to define a notion of "geometry." We will examine two such notions: (1) smooth manifolds with Riemannian metrics and (2) piecewise linear manifolds with piecewise flat structures. The former is a smooth manifold with a smoothly varying inner product at each point, while the latter is a collection of Euclidean simplices glued together along their boundaries. It turns out that in each case, we have a notion of a total curvature associated to the geometry, giving a functional (the Einstein-Hilbert functional) on the space of all such geometries. Critical points of these functionals give canonical geometries (many of which we all know and love, like the sphere). We will study this process and give some ideas of how one might try to find canonical geometries using such functionals. We will see connections with a number of well-studied areas of geometry, especially the Yamabe Problem, the Regge calculus, and Thurston's theory of discrete conformal maps via circle packings. However, I will not assume the audience has any previous knowledge of any of these areas.