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 $bf{hat{z}} times bf{E}$, where $bf{hat{z}}$ is the interface normal and $bf{E}$ denotes the electric field. The Co/Cu and Co/Pt interfaces additionally generate spin currents with polarization along $bf{hat{m}} times (bf{hat{z}} times bf{E})$, where $bf{hat{m}}$ 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.