Abstract. While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group and an arbitrary representation R of in GL(n, C), their existence, as far as we know, has been established in the literature only when restrictions are imposed on or R. In this paper, we prove the existence of n linearly independent vector-valued automorphic forms for any Fuchsian group and any n-dimensional complex representation R of . 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 non-compact Riemann surfaces and Kodaira's vanishing theorem.