We generalize Wertheim's 1-R renormalized perturbation theory for polar, polarizable fluids to mixtures in which the molecules possess dipoles and quadrupoles. The theory is used to calculate thermodynamic excess functions and phase equilibria for a variety of model mixtures, and comparisons with experiment are made for three binary mixtures. The results demonstrate the importance of accounting for many-body induction effects and for anisotropy of the polarizability. The additional induction effects due to the presnce of the quadrupole are appreciable provided one or more of the molecular species in the mixture possesses a dipole, but are small otherwise.