Trefftz variational methods, originally introduced by Cessenat and Despres , are discontinuous Galerkin numerical methods whose basis functions are solutions of the underlying partial differential equation that we want to solve numerically. They benefit from a solid theoretical framework that ensures their convergence. These methods can be solved iteratively and define a domain decomposition method. This drastically reduce their memory cost avoiding to resort to a LU decomposition. However, they are polluted by rounding errors that have severely limited their use in 3D. In this presentation, I will provide a new perspective to these methods in the context of linear hyperbolic problems. This includes a large variety of PDE systems like heterogeneous and anisotropic acoustic, elastic, and Maxwell systems. Recalling the theory of Friedrichs and Rauch, I will explain how to define general boundary conditions for this type of boundary value problems. Finally, I will explain how these variational methods can be modified to limit the impact of rounding errors. I will present two alternative techniques: a filtering method and a modification of the basis functions known as quasi-Trefftz. The presentation will conclude with concrete illustrations on very large computational scenes.