  The main purpose of the present work is an investigation of the notions
Hopfian (co-Hopfian) acts whose their surjective (injective) endomorphisms are
isomorphisms. While we investigate conditions that are relevant to these
classes of acts, their interrelationship with some other concepts for example
quasi-injective and Dedekind-finite acts is studied. Using Hopfian and
co-Hopfian concepts, several conditions are given for a quasi-injective act to
be Dedekind-finite. Moreover we bring out some properties of strongly Hopfian
and strongly co-Hopfian $S$-acts. Ultimately we introduce and study the concept
of Fitting acts and over a monoid $S$, some equivalent conditions are found to
have all its finitely generated (cyclic) acts Fitting. It is shown that an
$S$-act is Fitting if and only if it is both strongly Hopfian and strongly
co-Hopfian.
