  We prove that if $X = X_1 \times \dots \times X_n$ is a product of hyperbolic
Riemann surfaces of finite type and $Y = \Omega/\Gamma$ is a complex manifold,
where $\Omega$ is a bounded simply-connected domain in $\mathbb{C}^m$, then the
space of dominant holomorphic mappings from $X$ to $Y$ is a finite set. As
corollaries, we obtain the finiteness of the space of dominant holomorphic
mappings into products and symmetric products of hyperbolic Riemann surfaces.
