We develop stable and efficient Petrov-Galerkin discretizations for two transport-dominated problems: first order linear transport equations and kinetic Fokker-Planck equations. Based on well-posed weak formulations we first choose a discrete test space for the Petrov-Galerkin projection. A problem-dependent discrete trial space is then computed such that the spaces consist of matching stable pairs of trial and test functions. Thereby we obtain efficiently computable and uniformly inf-sup stable discrete schemes. For parametrized transport equations, we apply the reduced basis method and build a reduced model consisting of a fixed reduced test space and parameter-dependent reduced trial spaces depending on the test space. Due to the inherent stability we can avoid additional stabilizations in the basis generation so that we obtain efficient reduced models by an easily implemented procedure.

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