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A rich literature on the analysis of efficiency in production has developed since pioneering work of Tjalling Koopmans and George Debreu in the 1950s. This literature includes work by researchers in economics, econometrics, management science, operations research, mathematical statistics and other fields. The focus in this survey is on nonparametric approaches for estimation and inference, with particular emphasis on inference since nothing can be learned from estimation without inference. The statistical problem amounts to estimating the support of a multivariate random variable, subject to some shape constraints, in multiple dimensions. New results that enable inference about mean efficiency as well as tests of convexity, returns to scale, differences in mean efficiency and the separability condition described by Simar and Wilson (Journal of Econometrics 136:31–64, 2007) are discussed. The well-known curse of dimensionality presents additional challenges, but recent work indicates that reducing dimensionality using eigensystem techniques may improve estimation accuracy. Remaining challenges and open issues are also discussed.