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(2m+1)).　By　analyzing　the　supercloseness　between　the　DG　solution　and　the　interpolation　Pi(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.　(C)　2018　Elsevier　B.V.　All　rights　reserved.