Strong external difference families (SEDFs) have applications to cryptography and are rich combinatorial structures in their own right. We extend the definition of SEDF from abelian groups to all finite groups, and introduce the concept of equivalence. We prove new recursive constructions for SEDFs and generalized SEDFs (GSEDFs) in cyclic groups, and present the first family of non-abelian SEDFs. We prove there exist at least two non-equivalent (k² + 1,2,k,1)-SEDFs for every k > 2, and begin the task of enumerating SEDFs, via a computational approach which yields complete results for all groups up to order 24.