  We introduce the {\it endomorphism distinguishing number} $D_e(G)$ of a graph
$G$ as the least cardinal $d$ such that $G$ has a vertex coloring with $d$
colors that is only preserved by the trivial endomorphism. This generalizes the
notion of the distinguishing number $D(G)$ of a graph $G$, which is defined for
automorphisms instead of endomorphisms.
  As the number of endomorphisms can vastly exceed the number of automorphisms,
the new concept opens challenging problems, several of which are presented
here.
  In particular, we investigate relationships between $D_e(G)$ and the
endomorphism motion of a graph $G$, that is, the least possible number of
vertices moved by a nontrivial endomorphism of $G$. Moreover, we extend
numerous results about the distinguishing number of finite and infinite graphs
to the endomorphism distinguishing number. This is the main concern of the
paper.
