  A topological monoid is isomorphic to an endomorphism monoid of a countable
structure if and only if it is separable and has a compatible complete
ultrametric such that composition from the left is non-expansive. We also give
a topological characterisation of those topological monoids that are isomorphic
to endomorphism monoids of countable omega-categorical structures. Finally we
present analogous characterisations for polymorphism clones of countable
structures and for polymorphism clones of countable omega-categorical
structures.
