Typically, such a “structural” approach incorporates two types of assumptions: (a) economic assumptions, such as behavioral assumptions (e.g. Bayesian Nash equilibrium) and economically motivated restrictions on preferences (e.g., risk neutrality), and (b) functional form assumptions, imposed either for convenience in estimation or because only a limited set of parameters can be identified. An attractive feature of the recent econometric literature on auctions is that often the second type of assumption can be avoided, both in principle and in practice. In particular, in many cases identification of economic primitives can be obtained without resorting unverifiable parametric assumptions, and nonparametric estimation methods have been developed that perform well in data sets of moderate size. Even when parametric estimation approaches are used in applications, the fact that the literature has provided definitive positive (and sometimes negative) identification results provides important guidance about how to interpret the results. This paper aims to review some of the highlights of this recent literature, focusing on econometric identification and empirical applications.