Robust Bayesian Analysis Robust Bayesian analysis aims at overcoming the traditional objection to Bayesian analysis of its dependence on subjective inputs mainly the prior and the loss. Its purpose is the determination of the impact of the inputs to a Bayesian analysis (the prior the loss and the model) on its output when the inputs range in certain classes. If the impact is considerable there is sensitivity and we should attempt to further refine the information the incumbent classes available perhaps through additional constraints on and or obtaining additional data; if the impact is not important robustness holds and no further analysis and refinement would be required. Robust Bayesian analysis has been widely accepted by Bayesian statisticians; for a while it was even a main research topic in the field. However to a great extent their impact is yet to be seen in applied settings. This volume therefore presents an overview of the current state of robust Bayesian methods and their applications and identifies topics of further in­ terest in the area. The papers in the volume are divided into nine parts covering the main aspects of the field. The first one provides an overview of Bayesian robustness at a non technical level. The paper in Part II con­ cerns foundational aspects and describes decision theoretical axiomatisa­ tions leading to the robust Bayesian paradigm motivating reasons for which robust analysis is practically unavoidable within Bayesian analysis. 1 Bayesian Robustness 2 Topics on the Foundations of Robust Bayesian Analysis 3 Global Bayesian Robustness for Some Classes of Prior Distributions 4 Local Robustness in Bayesian Analysis 5 Global and Local Robustness Approaches: Uses and Limitations 6 On the Use of the Concentration Function in Bayesian Robustness 7 Likelihood Robustness 8 Ranges of Posterior Expected Losses and —Robust Actions 9 Computing Efficient Sets in Bayesian Decision Problems 10 Stability of Bayes Decisions and Applications 11 Robustness Issues in Bayesian Model Selection 12 Bayesian Robustness and Bayesian Nonparametrics 13  Minimax: A Paradigm for Conservative Robust Bayesians 14 Linearization Techniques in Bayesian Robustness 15 Methods for Global Prior Robustness under Generalized Moment Conditions 16 Efficient MCMC Schemes for Robust Model Extensions Using Encompassing Dirichlet Process Mixture Models 17 Sensitivity Analysis in IctNeo 18 Sensitivity of Replacement Priorities for Gas Pipeline Maintenance 19 Robust Bayesian Analysis in Medical and Epidemiological Settings 20 A Robust Version of the Dynamic Linear Model with an Economic Application 21 Prior Robustness in Some Common Types of Software Reliability Model.