Bayesian inference is a popular and practical tool for statistical inference. However, practitioners must make two major modeling choices when using Bayesian inference: the choice of the prior and likelihood. Uncertainty in these choices gives rise to the study of Bayesian robustness, which in part seeks to answer how posterior inferences would change had a practitioner made different modeling choices. I will give an overview of the field of Bayesian robustness with an emphasis on sensitivity to the specification of the prior and sensitivity to likelihood misspecification. To highlight practical implications of these issues, I will give some examples of how standard Bayesian inference for widely used models such as linear regression and Gaussian mixtures models can be dangerously non-robust.