The renormalization group running of fermion mixing matrices in the Standard Model and beyond is studied. For the massless one-loop running with three generations, six fixed points are found. Their associated anomalous dimension matrices are calculated and the nature of each fixed point, whether attractive, repulsive, or mixed, is determined. An argument is given that the fixed points found at one loop must remain fixed points to all orders in perturbation theory and even nonperturbatively, as they are associated with certain differential geometric properties of vector fields on the space of mixing matrices. With $N_g$ dark or sterile neutrinos there are at least $N_g!$ fixed points of the fermion mixing matrix.