The present paper is based essentially upon $R$. Nevanlinna's theory [I] on Abelian integrals. In § 1 we shall establish the first fundamental theorem of R. Nevanlinna (Theorem 2) for the meromorphic covariants of order $n (n=0, \pm 1, \pm 2, \ldots )$ (cf. sec. 4) on an arbitrary open Riemann surface and give some of its applications in § 2, where Theorem 3 corresponds to Shimizu-Ahlfors’ theorem for covariants.