In this article we develop robust indirect inference for a variety of models in a unified framework. We investigate the local robustness properties of indirect inference and derive the influence function of the indirect estimator, as well as the level and power influence functions of indirect tests. These tools are then used to design indirect inference procedures that are stable in the presence of small deviations from the assumed model. Although indirect inference was originally proposed for statistical models whose likelihood is difficult or even impossible to compute and/or to maximize, we use it here as a device to robustify the estimators and tests for models where this is not possible or is difficult with classical techniques such as M estimators. Examples from financial applications, time series, and spatial statistics are used for illustration.