Conventional mammography is one of the most widely used diagnostic imaging techniques, but it has serious and well-known shortcomings, which are driving the development of innovative alternatives. Our group has been developing an x-ray imaging approach called multiple-image radiography (MIR), which shows promise as a potential alternative to conventional x-ray imaging (radiography). Like computed tomography (CT), MIR is a computed imaging technique, in which the images are not directly observed, but rather computed algorithmically. Whereas conventional radiography produces just one image depicting absorption effects, MIR simultaneously produces three images, showing separately the effects of absorption, refraction, and ultra-small-angle x-ray scattering. The latter two effects are caused by refractive-index variations in the object, which yield fine image details not seen in standard radiographs. MIR has the added benefits of dramatically lessening radiation dose, virtually eliminating scatter degradation, and lessening the importance of compressing the breast during imaging. In this paper we review progress to date on the MIR technique, focus on the basic physics and signal-processing issues involved in this new imaging method.