In our talk we present a new robust, accurate and very simple a posteriori subcell finite volume limiter technique for the Discontinuous Galerkin (DG) finite element method for nonlinear systems of hyperbolic partial differential equations in multiple space dimensions that works well for arbitrary high order of accuracy in space and time and that does not destroy the natural subcell resolution properties of the DG method. High order time discretization is achieved via a fully-discrete one-step ADER approach that uses a local space-time discontinuous Galerkin predictor method to evolve the data locally in time within each cell. The new limiting strategy is based on a novel a posteriori verification of the validity of a discrete candidate solution against physical and numerical detection criteria. In particular, we employ a relaxed discrete maximum principle, the positivity of the numerical solution and the absence of floating point errors as detection criteria. For those troubled cells that need limiting, our new approach recomputes the discrete solution by starting again from a valid solution at the old time level, but using a more robust finite volume scheme on a refined subgrid of N_s=2N+1 subcells, where N is the polynomial approximation degree of the DG scheme. The new method can be interpreted as an element-local check-pointing and restarting of the solver, but using a more robust scheme on a finer mesh after the restart. The performance of the new method is shown on a large set of different hyperbolic partial differential equations systems using uniform and space-time adaptive Cartesian grids (AMR), as well as on unstructured meshes in two and three space dimensions. In particular, we will also show applications to a new unified first order hyperbolic theory of continuum mechanics proposed by Godunov, Peshkov & Romenski (GPR model). The presented research was financed by the European Research Council (ERC) with the research project STiMulUs, ERC Grant agreement no. 278267 and by the European Union's Horizon 2020 Research and Innovation Programme under the project ExaHyPE, Grant agreement number No.671698 (call FETHPC-1-2014).