Functional brain activity mainly occurs on the surface of the human brain in the grey matter. Interestingly, this surface is topologically equivalent to a two-dimensional sheet. As a result, it is possible to flatten and unfold this surface to create a flat map of the human brain. This map can then be used to map and localize functional activity. In this seminar, I will present a novel computer realization of the Riemann Mapping Theorem that uses circle packings to create quasi-conformal flat maps. I will discuss some of the topological difficulties encountered while trying to obtain a surface representation of the brain from high-resolution 3D magnetic resonance (MRI) images and present some of the quasi-conformal maps I have created in the Euclidean plane, in the hyperbolic plane and on a sphere.