  We compute the Mori cone of curves of the moduli space \M_{g,n} of stable
n-pointed curves of genus g in the case when g and n are relatively small. For
instance, we show that for g<14 every curve in \M_g is numerically equivalent
to an effective sum of 1-strata (loci of curves with 3g-4 nodes). We also prove
that the nef cone of \M_{0,6} is composed of 11 natural subcones all contained
in the convex hull of boundary classes. We apply this result to classify the
fibrations of the moduli space of rational curves with n<7 marked points.
