Magnetic quadrupole-octopole systems are treated theoretically. A technique is given for finding the minimum octopole field that will correct any geometrical aberration of any orthogonal quadrupole system. It is demonstrated with the correction of aperture aberration and coma in anamorphotic and orthomorphic objective type systems. Further minimization of octopole strengths is achieved by manipulating the quadrupole fields in systems corrected for aperture aberrations. Octopole strengths vary by two orders of magnitude in the cases studied, but chromatic aberrations become large when the octopoles are weakened. Quadrupole-octopole devices that correct the spherical aberration of round lenses are briefly considered. The antisymmetric corrector with minimum fields is very similar to the Archard-Deltrap corrector. It should be at least seven times longer than the focal length of the round lens in order to have fields and chromatic aberrations smaller than those of the best multipole systems without round lenses.