  The purpose of this paper is to introduce the theory of presentations of
monoids acts. We aim to construct `nice' general presentations for various act
constructions pertaining to subacts and Rees quotients. More precisely, given
an $M$-act $A$ and a subact $B$ of $A$, on the one hand we construct
presentations for $B$ and the Rees quotient $A/B$ using a presentation for $A$,
and on the other hand we derive a presentation for $A$ from presentations for
$B$ and $A/B$. We also construct a general presentation for the union of two
subacts. From our general presentations, we deduce a number of finite
presentability results. Finally, we consider the case where a subact $B$ has
finite complement in an $M$-act $A$. We show that if $M$ is a finitely
generated monoid and $B$ is finitely presented, then $A$ is finitely presented.
We also show that if $M$ belongs to a wide class of monoids, including all
finitely presented monoids, then the converse also holds.
