One of the main reasons to be interested in a.e. approximately differentiable mappings is the fact proved by Federer that they are admissible for area formula. I will talk about properties of such maps and focus on the question of prescribing the derivative of approximately differentiable homeomorphisms and how it is connected with Oxtoby-Ulam theorem about homeomorphic measures. I will show a stronger version of the theorem presented in the fall: that under some mild assumptions on a map T defined on the unit cube Q with values in GL(n), we can find an approximately differentiable homeomorphism, which equals identity on the boundary of Q and whose approximate derivative equals T a.e. This is joint work with Paweł Goldstein and Piotr Hajłasz.