Nonparametric Bayesian models provide the flexibility needed to match the richness of modern data. They allow one to capture nonlinearities in regressions, changes in distributional form, variation in the degree of local flexibility, and complex dependence structures. The models are inherently infinite dimensional and so behave differently than low-dimensional parametric models. While the flexibility provided by these models allows them to uncover patterns in the data, this same flexibility makes them prone to overfitting-pursuing idiosyncrasies in a given data set that neither reflect underlying ``truth'' nor will be replicated in future data sets. The methods often exhibit excellent robustness at a global level but show a lack of robustness in some locales. A variety of strategies exist for improving the robustness of the methods, ranging from techniques for prior specification to constrain the flexibility of the models, to choice of the likelihood, to the mechanism for producing inference from the model. This talk will focus on mechanisms for enhancing robustness of nonparametric Bayesian methods.