The study of metrics and probability distributions on smooth shapes (curves and surfaces) has recently become an active area of research, with applications in computer vision and computational anatomy, among others. I’ll give an overview of some current approaches used in shape representation and then focus specifically on one approach rooted in Teichmuller theory, where the isomorphism between Diff(S1)/PSL2(R) and (simple closed plane curves)/(translation,scale) provides a convenient representation for smooth shapes. The underlying conformal maps encode a sort of continuous version of the medial axis of a planar domain; conversely, study of the medial axis provides explicit estimates for invariant quantities related to conformal and hyperbolic structures on smooth domains. I’ll also discuss a family of paths which join smooth shapes to the unit disk. One path is given by an ODE which generates the isometry between the convex hull boundary and the hyperbolic plane; another related path is constructed using coupled Loewner evolutions.