There have been many theoretical studies and numerical investigations of nonlocal diffusion(ND)problems in recent years.In this paper,we propose and analyze a new dis-continuous Galerkin method for solving one-dimensional steady-state and time-dependent ND problems,based on a formulation that directly penalizes the jumps across the element interfaces in the nonlocal sense.We show that the proposed discontinuous Galerkin scheme is stable and convergent.Moreover,the local limit of such DG scheme recovers classical DG scheme for the corresponding local diffusion problem,which is a distinct feature of the new formulation and assures the asymptotic compatibility of the discretization.Numerical tests are also presented to demonstrate the effectiveness and the robustness of the proposed method.