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The fundamental problem of distance geometry involves the characterization and study of sets of points based only on given values of some or all of the distances between pairs of points. This problem has a wide range of applications in various areas of mathematics, physics, chemistry, and engineering. Euclidean distance matrices play an important role in this context by providing elegant and powerful convex relaxations. They play an important role in problems such as graph realization and graph rigidity. Moreover, by relaxing the embedding dimension restriction, these matrices can be used to approximate the hard problems efficiently using semidefinite programming. Throughout this survey we emphasize the interplay between these concepts and problems. In addition, we illustrate this interplay in the context of the sensor network localization problem.