Homeomorphisms with a derivative have become interesting from the point of view of studying deformations of materials (e.g., stretching a rubber band). Especially J. M. Ball’s nonlinear elasticity theory introduced many questions about the interplay of topological and differential properties of homeomorphisms. I will talk about approximately differentiable homeomorphisms between Euclidean subsets; the properties of their derivatives and Jacobians. In particular, I will show that a measurable mapping, satisfying some mild conditions, can be the approximate derivative of a homeomorphism of the unit cube. I will also discuss the implications of this result for the possibility of using this class of mappings in nonlinear elasticity. Based on a joint work with Paweł Goldstein (University of Warsaw) and Piotr Hajłasz (University of Pittsburgh).