We study the full discretization of the elliptic and parabolic $p$-Laplacian using discontinuous Galerkin methods. First, we consider an $hp$-version robust interior penalty discontinuous Galerkin method for the elliptic $p$-Laplacian on simplicial, as well as essentially arbitrarily-shaped polygonal and curved meshes. By employing novel quasi-norm trace–inverse estimates, we establish quasi-norm error bounds of optimal order with respect to the local mesh sizes and of slightly reduced order with respect to the local polynomial degrees. Moreover, the analogous results for a space-time discontinuous Galerkin formulation of the parabolic $p$-Laplacian will be discussed. Numerical experiments are presented to validate and illustrate the theoretical results. This is based on joint work with Konstantinos Chrysafinos (NTU Athens & IACM, FORTH) and Emmanuil H. Georgoulis (NTU Athens & Heriot-Watt Uni. & IACM, FORTH).