Bayesian methods for the analysis of clinical trials data have received increasing attention recently as they offer an approach for dealing with difficult problems that arise in practice. A major criticism of the Bayesian approach, however, has focused on the need to specify a single, often subjective, prior distribution for the parameters of interest. In an attempt to address this criticism, we describe methods for assessing the robustness of the posterior distribution to the specification of the prior. The robust Bayesian approach to data analysis replaces the prior distribution with a class of prior distributions and investigates how the inferences might change as the prior varies over this class. The purpose of this paper is to illustrate the application of robust Bayesian methods to the analysis of clinical trials data. Using two examples of clinical trials taken from the literature, we illustrate how to use these methods to help a data monitoring committee decide whether or not to stop a trial early.