  While vector-valued automorphic forms can be defined for an arbitrary
Fuchsian group $\Gamma$ and an arbitrary representation $R$ of $\Gamma$ in
GL$(n,{\mathbb C})$, their existence has been established in the literature
only when restrictions are imposed on both $\Gamma$ and $R$. In this paper, we
prove the existence of $n$ linearly independent vector-valued automorphic forms
for any Fuchsian group $\Gamma$ and any $n$-dimensional complex representation
$R$ of $\Gamma$. To this end, we realize these automorphic forms as global
sections of a special rank $n$ vector bundle built using solutions to the
Riemann-Hilbert problem over various noncompact Riemann surfaces and Kodaira's
vanishing theorem.
