The paper develops estimation and inference methods for econometric models with partial identification, focusing on models defined by moment inequalities and equalities. Main applications of this framework include analysis of game-theoretic models, revealed preference, regression with missing and mismeasured data, auction models, bounds in structural quantile models, bounds in asset pricing, among many others. Specifically, this paper provides estimators and confidence regions for minima of an econometric criterion function Q([Theta]). In applications, Q([Theta]) embodies testable restrictions on economic models. A parameter [Theta] that describes an economic model passes these restrictions if Q([Theta]) attains the minimum value normalized to be zero. The interest therefore focuses on the set of parameters [Theta]I that minimizes Qn([Theta]), called the identified set. This paper uses the inversion of the sample analog Q([Theta]) of the population criterion Q([Theta]) to construct the estimators and confidence regions for [Theta]I. We develop consistency, rates of convergence, and inference results for these estimators and regions. The results are shown to hold under general yet simple conditions, and practical procedures are provided to implement the approach. (cont.) In order to derive these results, the paper also develops methods for analyzing the asymptotics of sample criterion functions under set identification. Keywords: Set estimator, contour sets, moment inequalities, moment equalities. JEL Classifications: C13, C14, C21, C41, C51, C53.