  Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$
has genus $g\geq 6$ and $Y$ has genus at most $2g-1$; in addition, suppose that
$Y$ is not closed if it has genus $2g-1$. Our main result asserts that every
non-trivial homomorphism $\Map(X) \to \Map(Y)$ is induced by an {\em
embedding}, i.e. a combination of forgetting punctures, deleting boundary
components and subsurface embeddings. In particular, if $X$ has no boundary
then every non-trivial endomorphism $\Map(X)\to\Map(X)$ is in fact an
isomorphism. As an application of our main theorem we obtain that, under the
same hypotheses on genus, if $X$ and $Y$ have finite analytic type then every
non-constant holomorphic map $\CM(X)\to\CM(Y)$ between the corresponding moduli
spaces is a forgetful map. In particular, there are no such holomorphic maps
unless $X$ and $Y$ have the same genus and $Y$ has at most as many marked
points as $X$.
