We introduce a space-time discretization for elastic and acoustic waves using a discontinuous Galerkin approximation in space, and a Petrov-Galerkin scheme in time. For the DG method, the upwind flux is evaluated by explicitly solving a Riemann problem. Then we show well-posedness and convergence of the discrete system. Based on goal-oriented dual-weighted error estimation, an adaptive strategy is introduced. The full space-time linear system is solved with a parallel multilevel preconditioner. Numerical experiments for acoustic and elastic waves underline the efficiency of the overall adaptive solution process.