The study of 2D shapes and their similarities is a central problem in the field of vision. One of the issues is finding a suitable complete representation of the space of all 2D-shapes and finding a metric space structure on this representation space. I'll talk about one representation introduced in the paper "2D-Shape Analysis Using Conformal Mapping" by E. Sharon and D. Mumford. If time permits, I'll talk about the Weil-Petersson Riemannian Metric defined on this space. The talk will be pretty much accessible to everyone (all you'll need to know are basics of conformal mappings and Mobius Transformations).