There are just five Fraïssé classes of permutations (apart from the trivial class of permutations of a singleton set); these are the identity permutations, reversing permutations, composites (in either order) of these two classes, and all permutations. The paper also discusses infinite generalisations of permutations, and the connection with Fraïssé's theory of countable homogeneous structures, and states a few open problems. Links with enumeration results, and the analogous result for circular permutations, are also described.