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. Classical results such as the Beurling-Ahlfors extension result or the Radó-Kneser-Choquet theorem constitute some of the basic building blocks needed to solve these problems in 2D space. In this talk, we will review the known planar theory of questions such as the Sobolev Jordan-Schönflies problem of extending a boundary map between Jordan domains as a Sobolev homeomorphism. Moreover, we discuss some first approaches in higher dimensions where many of the techniques crucial to the planar theory simply fail. This talk is based on joint work with Stanislav Hencl and Jani Onninen.