Strong external difference families (SEDFs) have applications to cryptography and are rich combinatorial structures in their own right; until now, all SEDFs have been in abelian groups. In this paper, we consider SEDFs in both abelian and non-abelian groups. We characterize the order of groups possessing admissible parameters for non-trivial SEDFs, develop non-existence and existence results, several of which extend known results, and present the first family of non-abelian SEDFs. We introduce the concept of equivalence for EDFs and SEDFs, and begin the task of enumerating SEDFs. Complete results are presented for all groups up to order $24$, underpinned by a computational approach.