  In this paper, we consider semigroup actions of discrete countable semigroups
on compact spaces by surjective local homeomorphisms. We introduce notions of
continuous one-sided orbit equivalence and continuous orbit equivalence for
semigroup actions, and characterize them in terms of the corresponding
semi-groupoids and transformation groupoids respectively. Finally, we consider
the case of semigroup actions by homeomorphisms and relate continuous orbit
equivalence of semigroup actions to that of group actions.
