Transport calculations based on ab initio band structures reveal large interface-generated spin currents at Co/Pt, Co/Cu, and Pt/Cu interfaces. These spin currents are driven by in-plane electric fields but flow out of plane and can have similar strengths to spin currents generated by the spin Hall effect in bulk Pt. Each interface generates spin currents with polarization along z[over ^]×E, where z[over ^] is the interface normal and E denotes the electric field. The Co/Cu and Co/Pt interfaces additionally generate spin currents with polarization along m[over ^]×(z[over ^]×E), where m[over ^] gives the magnetization direction of Co. The latter spin polarization is controlled by-but not aligned with-the magnetization, providing a novel mechanism for generating spin torques in magnetic trilayers.