Cayley's theorem in elementary group theory shows that every abstract group G may be represented as a concrete group of automorphisms, namely of G itself as a right G-set. A formally identical proof works for a family of structures and all isomorphisms between them, which now form a "multi-pointed" generalization of a group called a groupoid (and the representation theorem for them now known as the Yoneda lemma). Generalizing Cayley's theorem in another direction, every two-sided complete topological group can be represented as the automorphism group of some first-order structure (discrete if the group is non-Archimedean; metric in general). In this talk, we will explain these three related results, and their common generalization to a representation theorem for topological groupoids.