X-ray scattering, including coherent, incoherent, and resonance (fluorescent) phenomena, is of fundamental importance in contemporary science as a tool to probe the structure of matter at an atomic level. This elevated status is in sharp contrast to the role of x-ray scatter in medical and industrial radiography, where it is generally regarded as an unmitigated nuisance to be corrected for or, preferably, eliminated. We introduce a novel x-radiographic technique [x-ray diffraction computed tomography (CT)] based on measurement of coherent scatter. The physical background to coherent scattering is described in a simple, classical way, and its importance in conventional radiography is demonstrated by using Monte Carlo analysis. Some diffraction patterns of plastics and animal tissues are presented to illustrate the relevance of coherent scattering to material characterization and diagnosis in industrial and medical radiology. An experimental first-generation (single-pencil-beam) CT system has been constructed to demonstrate the feasibility of x-ray diffraction CT. Some diffraction CT and coherent scatter projection images of simple objects are presented. Possibilities for improving the ratio of image contrast to noise and increasing the momentum resolution of the technique are illustrated.