### Abstract.

A permutation group on a countably infinite domain is called *oligomorphic* if it has finitely many orbits of finitary tuples. We define a clone on a countable domain to be *oligomorphic* if its set of permutations forms an oligomorphic permutation group. There is a close relationship to *ω-categorical structures*, i.e., countably infinite structures with a first-order theory that has only one countable model, up to isomorphism. Every locally closed oligomorphic permutation group is the automorphism group of an *ω*-categorical structure, and conversely, the canonical structure of an oligomorphic permutation group is an *ω*-categorical structure that contains all first-order definable relations. There is a similar Galois connection between locally closed oligomorphic clones and *ω*-categorical structures containing all *primitive positive* definable relations.

In this article we generalise some fundamental theorems of universal algebra from clones over a finite domain to oligomorphic clones. First, we define *minimal* oligomorphic clones, and present equivalent characterisations of minimality, and then generalise Rosenberg’s five types classification to minimal oligomorphic clones. We also present a generalisation of the theorem of Baker and Pixley to oligomorphic clones.