Discontinuous Galerkin (DG) methods were proposed in 1973 to solve the neutron transport equation. In recent years they have become very popular with numerical analysts and engineers working on problems where the solution has a strong hyperbolic character. For example, for Maxwell's equations Hesthaven and Warburton have demonstrated very impressive numerical results for a method combining a discontinuous Galerkin method in space with the Runge-Kutta scheme in time.

In this talk we shall start by investigating the dispersion and dissipation behavior of a variety of discontinuous Galerkin formulations. It will turn out that the dispersive behavior of these methods is more complicated that for simple finite difference schemes. Finally we shall present a new space-time DG methods for symmetric hyperbolic systems, derive error estimates and show some preliminary numerical examples.