Conformal maps offer many nice mathematical properties for applications, including angle preservation, uniqueness, and the ability to create conformal maps in different geometries. Such features are important for medical applications, and in particular, conformal maps are being used to create maps of the brain. In this presentation, I will discuss how we create quasi-conformal "flat" maps of the brain from MRI brain scans using image segmentation, topology, and the computational method of circle packing. We are using these conformal brain maps to investigate the progression of brain diseases and illnesses, including Alzheimer's, schizophrenia and depression. Conformal invariants, such as extremal lengths, and their role in studying the shape of regions of the human brain will also be discussed.