Let $M_g^n$ be the moduli space of $n$-pointed Riemann surfaces of genus $g$. Denote by $\overline{M_g^n}$ the Deligne-Mumford compactification of $M_g^n$. In the present paper, we calculate the orbifold and the ordinary Euler characteristic of $\overline{M_g^n}$ for any $g$ and $n$ such that $n>2-2g$.