Digital x-ray tomosynthesis is a technique for producing slice images using conventional x-ray systems. It is a refinement of conventional geometric tomography, which has been known since the 1930s. In conventional geometric tomography, the x-ray tube and image receptor move in synchrony on opposite sides of the patient to produce a plane of structures in sharp focus at the plane containing the fulcrum of the motion; all other structures above and below the fulcrum plane are blurred and thus less visible in the resulting image. Tomosynthesis improves upon conventional geometric tomography in that it allows an arbitrary number of in-focus planes to be generated retrospectively from a sequence of projection radiographs that are acquired during a single motion of the x-ray tube. By shifting and adding these projection radiographs, specific planes may be reconstructed. This topical review describes the various reconstruction algorithms used to produce tomosynthesis images, as well as approaches used to minimize the residual blur from out-of-plane structures. Historical background and mathematical details are given for the various approaches described. Approaches for optimizing the tomosynthesis image are given. Applications of tomosynthesis to various clinical tasks, including angiography, chest imaging, mammography, dental imaging and orthopaedic imaging, are also described.