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This book gives a comprehensive and up-to-date account of the optimisation approach to shape correspondence, and the question of evaluating the quality of the resulting model in the absence of ground-truth data. It begins with a complete account of the basics of statistical shape models, for both finite and infinite-dimensional representations of shape, and includes linear, non-linear, and kernel-based approaches to modelling distributions of shapes. The optimisation approach is then developed, with a detailed discussion of the various objective functions available for establishing correspondence, and a particular focus on the Minimum Description Length approach. Various methods for the manipulation of correspondence for shape curves and surfaces are dealt with in detail, including recent advances such as the application of fluid-based methods.