one of the simplest model problems in the kinetic theory of plasma--physics is the vlasov-poisson system with periodic boundary conditions. such system describes the evolution of a plasma of charged particles (electrons and ions) under the effects of the transport and self-consistent electric field. in this talk, we present some discontinuous galerkin (dg) methods for the approximation of the vlasov-poisson system. the schemes are based on the coupling of dg approximation to the vlasov equation (transport equation) and several finite element (conforming, non-conforming and mixed) approximations to the poisson problem. we present the error analysis and discuss further properties of the proposed schemes. we also present numerical experiments in the 1d case that verify the theory and validate the performance of the methods in benchmark problems. if time allows, in the last part of the talk, we shall discuss the possibility of combining the proposed methods with some dimension reduction techniques, such as sparse grids. the talk is based on joint works with saverio castelanelli (uab-crm), j.a. carrillo (imperial college-icrea), soheil hajian (univ. geneva) and chi-wang shu (brown university).