The human brain is a highly convoluted surface with many folds and fissures that vary considerably from person to person, making it difficult to compare functional activation across individuals. Most of the functional activity of the brain occurs on the surface, called the "grey matter". Since this surface is topologically equivalent to a sheet, there is great interest in "unfolding" it to create a flat map of the brain. I will discuss a novel technique that we developed which uses the Riemann Mapping Theorem and circle packings to create quasi-conformal flat maps of the brain. These maps can be produced in the Euclidean and hyperbolic planes and on a sphere. I will present examples of these maps and show how they are being used to elucidate new information about the human brain.