Schemes for focusing a hard x-ray beam to a small spot are described. The theoretical minimum spot size, assuming perfect mirror shape, is shown to be $4\mathrm{nm}$ FWHM, independent of x-ray wavelength. This is less than the $10\mathrm{nm}$ previously said to be the minimum achievable diffraction-limited x-ray spot size. While providing the penetrating power only possible with x rays, this approaches the resolution needed to image individual atoms or atomic layers. However, the perfect mirror assumption is physically unrealistic. This paper discusses the compensation of mirror shape errors by a corrector plate and shows that the tolerances for corrector plate shape are far looser than are tolerances for mirror shape. The full eventual success of achieving theoretical minimum resolution will require mirror shape precision considerably better than has been achieved at this time, though far looser than would be required for simpleminded paraboloidal focusing. Two variants of the scheme, subject to the same mathematical treatment, are described. (i) The “corrector plate” name is copied from the similarly functioning element of the same name in a Schmidt camera. The focusing is achieved using glancing, yet coherent, reflection from a high-Z paraboloidal mirror. The strategy is to obtain dominant focusing from reflection and to compensate with weak refractive focusing. The reflective focusing is strong and achromatic but insufficiently accurate. The refractive focusing is weak and chromatic but highly accurate. The corrector plate improves resolution the way eyeglasses help a person to see. It can, for example, be “fitted” the same trial-and-error way an optometrist establishes a prescription for glasses. Dimensional tolerances for the compensator are far looser than would be needed for a mirror to achieve the same resolution. Unlike compound refractive lenses, attenuation will be small, at least for wavelengths longer than $1\hspace{0.17em}\AA$ , because the compensation layer is thin. (ii) For this variant, the corrector plate is a washer-shaped refractive or Fresnel lens, and the mirror is (theoretically) a perfect cone. All focusing is provided by the lens. Even though the cone provides no focusing, it improves the resolution by increasing the numerical aperture of the device. Compared to a paraboloidal shape, it is assumed that the conical shape can be more accurately fabricated. Of the two variants, only the first variant is, in principle, capable of achieving the theoretical minimum resolution. Configurations are suggested, in both case (i) and case (ii), that use currently possible construction precisions to produce resolutions better than have been achieved to date. However, both results will remain well above the theoretical minimum until fabrication techniques have been developed that provide greater precision than is possible at this time.