The author investigates the conditions under which some limiting solutions of the kinetic Vlasov-Poisson system governing the plasma motion can be described by the hydrodynamic solutions of Euler-Poisson system (the so-called “hydrodynamic limit”). A rich literature is devoted to the subject, and the novelty of this thesis is that the corresponding results are obtained for arbitrary Hölder continuous initial density and velocity. Additionally, the author shows that to any sufficiently smooth solution of the Euler-Poisson system corresponds a unique solution of the Vlasov-Poisson system (with a canonical distribution function). The proofs are mainly based on the Schauder-type estimates in Hölder norms and on a detailed study of the conservation laws and characteristics of the Vlasov-Poisson system.