  We prove that the moduli space of polarized $K3$ surfaces of genus eleven
with $n$ marked points is unirational when $n\leq 6$ and uniruled when
$n\leq7$. As a consequence, we settle a long standing but not proved assertion
about the unirationality of $\cal{M}_{11,n}$ for $n\leq6$. We also prove that
the moduli space of polarized $K3$ surfaces of genus eleven with $9$ marked
points has non-negative Kodaira dimension.
