The quotient of a real analytic manifold by a properly discontinuous group action is, in general, only a semianalytic variety. We study the boundary of such a quotient, i.e., the set of points at which the quotient is not analytic. We apply the results to the moduli space M$_g/R$ of nonsingular real algebraic curves of genus g (g[les ]2). This moduli space has a natural structure of a semianalytic variety. We determine the dimension of the boundary of any connected component of M$_g/R$. It turns out that every connected component has a nonempty boundary. In particular, no connected component of M$_g/R$is real analytic. We conclude that M$_g/R$ is not a real analytic variety.