In this paper, we investigate the local superconvergence of the discontinuous Galerkin (DG) solutions on quasi-graded meshes for nonlinear delay differential equations with vanishing delay. It is shown that the optimal order of the DG solution at the mesh points is O ( h 2 m + 1 ) . By analyzing the supercloseness between the DG solution and the interpolation Π h u of the exact solution, we get the optimal order O ( h m + 2 ) of the DG solution at characteristic points. We then extend the convergence results of DG solutions to state dependent delay differential equations. Numerical examples are provided to illustrate the theoretical results.