Flat maps of the human brain computed from magnetic resonance imaging (MRI) scans have increasingly been used in visualizing anatomical and functional data of the brain. However, flat maps have not been utilized to their full potential in terms of analysis or gaining insight into comparing healthy and diseased subject populations. I will discuss current flat mapping approaches and present results on using conformal, or angle-preserving, flat maps of the brain. Conformal maps offer a number of promising mathematical features that may be exploited to characterize shape. I will discuss some of these properties and recent results in using conformal maps to investigate major depressive disorder in the ventral medial prefrontal cortex.