This paper will concentrate on reviewing recent activity in the subject, of which there has been an explosion in recent years. The bulk of this recent work has been concerned with:
(i) modelling uncertainty in the prior by specifying a class, $\Gamma$, of possible prior distributions; and (ii) determining the range of the posterior quantity of interest as the prior ranges over $\Gamma$.
This can be thought of as an implementation of the `black box’ model for Bayesian robustness which was introduced by {\it I. J. Good} [see e.g. his discussion contribution to C. A. B. Smith, J. R. Stat. Soc., Ser. B 23, 1-37 (1961;

Zbl 0124.096)] and is discussed in Section 2. Section 3 presents a review of this material, comparing the strengths and weaknesses of the various methodologies that have been proposed, and illustrating the methodologies with numerical examples. Section 4 discusses one of the most immediate and important applications of robust Bayesian methodology, namely the calculation of lower bounds on Bayes factors in hypothesis testing.