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In a wide variety of applications it is necessary to infer the structure of a multidimensional object from a set of its projections. There has been a long-standing interest in this problem and a number of different techniques have been proposed. In this paper, we present a tutorial review of the reconstruction problem and some of the algorithms which have been proposed for its solution. In addition, we present a number of new algorithms that appear to have some advantages over previous algorithms. Some comparisons of these algorithms applied to reconstructions of two-dimensional pictures are given. Furthermore, a number of new theoretical results are presented relating to the minimum number of projections necessary for exact reconstruction.