  We prove that the moduli space ${\Cal M}_{g,n}$ of smooth curves of genus $g$
with $n$ marked points is rational for $g=6$ and $1 \le n \le 8$, and it is
unirational for $g=8$ and $1 \le n \le 11$, $g=10$ and $1 \le n \le 3$, $g=12$
and $n = 1$.
