This　paper　is　concerned　with　the　application　of　the　discontinuous　Galerkin　method　to　delay　differential　equations　with　vanishing　delay　qt　(0　＜　q　＜　1).　Our　aim　is　to　establish　optimal　global　and　local　superconvergence　results　on　uniform　meshes　and　compare　these　with　analogous　estimates　for　collocation　methods.　The　theoretical　results　are　illustrated　by　a　broad　range　of　numerical　examples.