Fraenkel and Mostowski developed a permutation-based model of set theory where the Axiom of Choice does not hold. This was around 1922-1938. Almost 100 years later, this simple model fostered a huge quantity of research results. Among these, we find Lawvere-style initial-algebra syntax with binders, final-coalgebra semantics with resource allocation, and minimization algorithms for mobile systems. These results are also guaranteed by describing FM-sets in terms of category theory, and proving an equivalence of models between several different categories. We aim at providing survey of some of these developments, framed in the recent research line of ``nominal computation theory'', where the essential notion of name is declined in several different ways.