  Let $\overline{\mathcal{M}}_{g,n}$ be the moduli stack parametrizing
Deligne-Mumford stable $n$-pointed genus $g$ curves and let
$\overline{M}_{g,n}$ be its coarse moduli space: the Deligne-Mumford
compactification of the moduli space of $n$-pointed genus $g$ smooth curves. We
prove that the automorphism groups of $\overline{\mathcal{M}}_{g,n}$ and
$\overline{M}_{g,n}$ are isomorphic to the symmetric group on $n$ elements
$S_{n}$ for any $g,n$ such that $2g-2+n\geq 3$, and compute the remaining
cases.
