We explore and compare a variety of definitions for privacy and disclosure limitation in statistical estimation and data analysis, including (approximate) differential privacy, testing-based definitions of privacy, and posterior guarantees on disclosure risk. We give equivalence results between the definitions, shedding light on the relationships between different formalisms for privacy. We also take an inferential perspective, where---building off of these definitions---we provide minimax risk bounds for several estimation problems, including mean estimation, estimation of the support of a distribution, and nonparametric density estimation. These bounds highlight the statistical consequences of different definitions of privacy and provide a second lens for evaluating the advantages and disadvantages of different techniques for disclosure limitation.