We　present　the　hp-version　of　the　discontinuous　Galerkin　method　for　the　numerical　solution　of　delay　differential　equations　with　nonlinear　vanishing　delays　and　derive　error　bounds　that　are　explicit　in　the　time　steps,　the　degrees　of　the　approximating　polynomials,　and　the　regularity　properties　of　the　exact　solutions.　It　is　shown　that　the　hp　discontinuous　Galerkin　method　exhibits　exponential　rates　of　convergence　for　smooth　solutions　on　uniform　meshes,　and　for　nonsmooth　solutions　on　geometrically　graded　meshes.　The　theoretical　results　are　illustrated　by　various　numerical　examples.