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We present a novel Discontinuous Galerkin Finite Element Method for wave propagation problems. The method employs space–time Trefftz basis functions that satisfy the underlying partial differential equations and the respective interface boundary conditions exactly in an element-wise fashion. The basis functions can be of arbitrarily high order and we demonstrate spectral convergence in the space–time L 2-norm. Therefore high order time integration is an inherent property of the method and clearly sets it apart from methods that employ a high order approximation in space only.