In this talk, I will introduce a filtering technique for Discontinuous Galerkin approximations of hyperbolic problems. After a brief overview of classical monotonization techniques, I will present an an approach which reduces the spurious oscillations that arise in presence of discontinuities when high order spatial discretizations are employed. This goal is achieved using a filter function that keeps the high order scheme when the solution is regular and switches to a monotone low order approximation if it is not, following an approach already proposed for the Hamilton-Jacobi equations by other authors. The method has been implemented in the framework of the deal.II numerical library, whose mesh adaptation capabilities are also used to reduce the region in which the low order approximation is used. The potentialities of the proposed filtering technique are shown in a number of numerical experiments.