In this paper,we consider the local discontinuous Galerkin method (LDG) for solving singularly perturbed convection-diffusion problems in one-and two-dimensional settings.The existence and uniqueness of the LDG solutions are verified.Numerical experiments demonstrate that it seems impossible to obtain uniform superconvergence for numerical fluxes under uniform meshes.Thanks to the implementation of two-type different anisotropic meshes,i.e.,the Shishkin and art improved grade meshes,the uniform 2p+1-order superconvergence is observed numerically for both one-dimensional and twodimensional cases.