Promoted by the successful application of multilayer coated optics in soft x-ray imaging experiments in solar physics and projection lithography, several groups have designed, analyzed, fabricated, and are testing Schwarzschild multilayer soft x-ray microscopes. Simulations have indicated that diffraction limited performance of a spherical Schwarzschild microscope operating near 100 Å will be limited to systems with a small numerical aperture of approximately 0.15 and a corresponding resolution, based on the Rayleigh criterion, of 3.3 times the wavelength of the incident radiation. In principle, a two aspherical mirror Head microscope, which satisfies the constant optical path length condition and the Abbé sine condition, should achieve diffraction limited performance for very large numerical apertures. For a practical soft x-ray microscope, surface contour errors, microroughness, reflectance of multilayer coatings, and variation of the angle of incidence over the multilayer substrates become significant factors in degrading system resolution and must be controlled before an ultra-high resolution, two-mirror microscope will be realized. For a 30x reflecting microscope with a numerical aperture ranging from 0.15 to 0.35, the effects on resolution of surface contour errors, tilts, and misalignments of the optics have been studied. Graded spacing of the multilayer coatings on the mirror substrates are required of a fast, two-mirror microscope. © 1995 IOS Press. All rights reserved.