This chapter focuses on the possibility of extending isomorphisms or isometries to maps of the same type. It presents all known results about the automorphic space problem of Lindenstrauss and Rosenthal, including a dichotomy theorem, and about spaces of universal disposition already envisioned by Gurariy. It also treats finite-dimensional variations of those properties: the rich theory of UFO spaces and finitely automorphic quasi-Banach spaces. The topics of how many positions a Banach space can occupy in a bigger superspace and how many twisted sums of two spaces exist are considered.