In the present paper we investigate geodesic mappings of manifolds with affine connection onto Ricci symmetric manifolds which are characterized by the covariantly constant Ricci tensor. We obtained a fundamental system for this problem in a form of a system of Cauchy type equations in covariant derivatives depending on no more than n(n + 1) real parameters. Analogous results are obtained for geodesic mappings of manifolds with afine connection onto symmetric manifolds.