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Absorption-contrast x-ray imaging serves to visualize the variation in x-ray attenuation within the volume of a given sample, whereas phase contrast allows one to visualize variations in x-ray refractive index. The former imaging mechanism has been well known and widely utilized since the time of Röntgen’s Nobel prize winning work, whereas the latter mechanism—sought for, but not found, by Röntgen himself—has laid the foundation for a revolution in x-ray imaging which is the central topic of this review. We consider the physical imaging mechanisms underlying both absorption contrast and phase contrast, together with the associated inverse problem of how one may obtain quantitative two-or three-dimensional information regarding a sample, given one or more phase-contrast images of the same. Practical questions are considered, regarding optimized phase-contrast imaging geometries as a function of …