Numerous experiments in a variety of applied disciplines involve measuring distances between pairs of objects. The statistical problem posed by such experiments is that of fitting the observed data with a model defined to be the Euclidean distances between an abstract configuration of points. Techniques for solving this problem are collectively known as multidimensional scaling. These techniques have a long history in psychometrics and multivariate statistics, a much shorter one in the application of distance geometry to problems of molecular conformation. This review attempts to integrate these two traditions, which presently exist almost unaware of each other. Emphasis is placed on the rigorous formulation of the defining optimization problems, and on the computational practices that have been developed for solving these problems. Recent developments suggest that multidimensional scaling has entered a new and exciting era, as researchers begin to apply the tools of modern computational mathematics.