The problem of optical design is stated as follows: Given a principal surface r($alpha$), and a maximum focal angle $alpha$/sub m/, find the pair of optical surfaces for which diffraction limited focusing is achieved. It is shown that specification of r($alpha$) and $alpha$/sub m/ uniquely determines the lens design to within a scale factor, given the refractive index of the lens. It is further shown that one straightforward Runge-Kutta integration routine generates both surfaces for either a lens or a pair of mirror surfaces. The complete family of aplanatic lenses is described. Deviation from sphericity is discussed, as well as the possibility of realizing the specified lens designs. The family of lenses which map uniform incident intensity into uniform illumination about the focus is also described. Extension of the method to off-axis aberrations is considered. (auth)