  A countable semigroup is $\aleph_0$-categorical if it can be characterised,
up to isomorphism, by its first-order properties. In this paper we continue our
investigation into the $\aleph_0$-categoricity of semigroups. Our main results
are a complete classification of $\aleph_0$-categorical orthodox completely
0-simple semigroups, and descriptions of the $\aleph_0$-categorical members of
certain classes of strong semilattices of semigroups.
