Circles, Graphs, Orbifolds

An “abstract polyhedron” means, roughly, a graph that “might be the edges and vertices of a polyhedron”.  When can we promote “might be” to “is”? This question is answered by a beautiful theorem about circle packings on the sphere. I will explain the proof of this theorem, as well as some historical quirks around its discovery and a connection to the (generalized) Riemann mapping theorem.