  In this paper we initiate the study of $\aleph_0$-categorical semigroups,
where a countable semigroup $S$ is $\aleph_0$-categorical if, for any natural
number $n$, the action of its group of automorphisms Aut $S$ on $S^n$ has only
finitely many orbits. We show that $\aleph_0$-categoricity transfers to certain
important substructures such as maximal subgroups and principal factors. We
examine the relationship between $\aleph_0$-categoricity and a number of
semigroup and monoid constructions, namely direct sums, 0-direct unions,
semidirect products and $\mathcal{P}$-semigroups. As a corollary, we determine
the $\aleph_0$-categoricity of an $E$-unitary inverse semigroup with finite
semilattice of idempotents in terms of that of the maximal group homomorphic
image.
