We review recent developments in the econometrics of shape restrictions and their role in applied work. Our objectives are threefold. First, we aim to emphasize the diversity of applications in which shape restrictions have played a fruitful role. Second, we intend to provide practitioners with an intuitive understanding of how shape restrictions impact the distribution of estimators and test statistics. Third, we aim to provide an overview of new advances in the theory of estimation and inference under shape restrictions. Throughout the review, we outline open questions and interesting directions for future research.

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