  In this paper we study the ample cone of the moduli space $\mgn$ of stable
$n$-pointed curves of genus $g$. Our motivating conjecture is that a divisor on
$\mgn$ is ample iff it has positive intersection with all 1-dimensional strata
(the components of the locus of curves with at least $3g+n-2$ nodes). This
translates into a simple conjectural description of the cone by linear
inequalities, and, as all the 1-strata are rational, includes the conjecture
that the Mori cone is polyhedral and generated by rational curves. Our main
result is that the conjecture holds iff it holds for $g=0$. More precisely,
there is a natural finite map $r: \vmgn 0. 2g+n. \to \mgn$ whose image is the
locus $\rgn$ of curves with all components rational. Any 1-strata either lies
in $\rgn$ or is numerically equivalent to a family $E$ of elliptic tails and we
show that a divisor $D$ is nef iff $D \cdot E \geq 0$ and $r^*(D)$ is nef. We
also give results on contractions (i.e. morphisms with connected fibers to
projective varieties) of $\mgn$ for $g \geq 1$ showing that any fibration
factors through a tautological one (given by forgetting points) and that the
exceptional locus of any birational contraction is contained in the boundary.
Finally, by more ad-hoc arguments, we prove the nefness of certain special
classes.
