One of the most fundamental problems in Geometric Analysis is to understand which properties of a boundary map allow for a homeomorphic extension with specific geometric and analytic properties. In Nonlinear Elasticity, the existence of a Sobolev homeomorphism between two given shapes constitutes an axiomatic property for the associated minimization problems to be well-defined. Classical results such as the Beurling-Ahlfors extension theorem or the Radó-Kneser-Choquet theorem provide the basic building blocks for the 2D case, but do not extend well to irregular domains or to higher dimensions. In this talk, I will review the methods developed by myself and coauthors such as Jani Onninen and Stanislav Hencl in recent years to address these problems. The main question I will consider is the Sobolev Jordan-Schönflies problem of extending a boundary map between planar Jordan domains as a Sobolev homeomorphism, and its highly challenging higher-dimensional alternatives.