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This work is about random measures stationary with respect to a possibly non-transitive group action. It contains chapters on Palm Theory, the Mass-Transport Principle and Ergodic Theory for such random measures. The thesis ends with discussions of several new models in Stochastic Geometry (Cox Delauney mosaics, isometry stationary random partitions on Riemannian manifolds). These make crucial use of the previously developed techniques and objects.