The spherical and chromatic aberrations of a converging electron mirror are of opposite sign from those of electron lenses. This important property makes it possible in principle to compensate the aberrations of electron lenses by means of an electron mirror and to design electron microscopes based on a corrected optics system incorporating an electron mirror. In this paper the properties of the hyperbolic electron mirror are calculated, and the conditions for simultaneous correction of spherical and chromatic aberrations are worked out for several types of electron microscopes. The hyperbolic mirror field is a rotationally symmetric potential field between two electrodes. The electrodes are shaped as equipotential surfaces of the hyperbolic field, except for an aperture on the axis of the positive electrode for entrance and exit of electrons. The effect of the aperture is to create a thin diverging aperture lens at the termination of the hyperbolic field. The properties of the mirror are calculated analytically. The problem of separating the electron beam incident on the mirror from the beam returning from the mirror without impairing the image quality is solved by means of magnetic deflecting fields located at image planes. The mirror corrections can be applied to either magnetic or electrostatic lenses. The parameters for correction of aberrations are calculated for systems using electrostatic lenses. With appropriate polarity of the accelerating voltage and the lens and mirror voltages the calculations apply to ion imaging systems as well.