  We show that certain geometrically defined higher codimension cycles are
extremal in the effective cone of the moduli space $\overline{\mathcal
M}_{g,n}$ of stable genus $g$ curves with $n$ ordered marked points. In
particular, we prove that codimension two boundary strata are extremal and
exhibit extremal boundary strata of higher codimension. We also show that the
locus of hyperelliptic curves with a marked Weierstrass point in
$\overline{\mathcal M}_{3,1}$ and the locus of hyperelliptic curves in
$\overline{\mathcal M}_4$ are extremal cycles. In addition, we exhibit
infinitely many extremal codimension two cycles in $\overline{\mathcal
M}_{1,n}$ for $n\geq 5$ and in $\overline{\mathcal M}_{2,n}$ for $n\geq 2$.
