This thesis aims at studying the local robustness properties of Bayesian posterior summaries and deriving a robust procedure to estimate Bayesian Mean-Variance weights in a portfolio selection problem. In the first part, we study the local robustness of Bayesian estimators. In particular, we build a framework wherein any Bayesian quantity can be seen as a posterior functional. In this way it becomes possible to construct different robustness measures. We derive local influence measures for posterior summaries with respect both to prior and sampling distributions and to observations. Then we address the issue of efficient implementation of the derived measures through MCMC algorithms. In the second part, we deal with the problem of robust estimation in a Bayesian context, providing a useful result to generalize univariate robust distributions to the multivariate case. We also propose criteria to assess in which cases a robust model is recommended and how to choose among estimates obtained with different distributions. Finally, we consider in the third part the Mean-Variance portfolio selection problem. We provide evidence that if the data are normally distributed the Bayesian approach works better than the Certainty Equivalence approach, nevertheless this is no longer true when the data contain few outlying observations. Moreover, we compute useful measures of sensitivity of Bayesian weights and we construct and implement a new estimator which is robust with respect to the presence of 'extreme' observations.