Riemann's mapping theorem is the gem of 19th century mathematics. It says that any simply-connected domain D of the complex plane C is conformally diffeomorphic to the standard open unit disk. This amounts to the uniformization theorem for surfaces. We will describe the Koebe-Andreev-Thurston realization theorem, and Thurston's discovery of the rigidity of hexagonal circle packings to prove Riemann's theorem. Lots of fun!