We start by defining the general notions of “risk” and “uncertainty” and by discussing the risk management process, in particular in a financial and insurance context. We see that robustness can be derived as a necessary property of risk management procedures from these definitions. In practice, however, regulatory requirements are of highest importance to insurance companies. Therefore, we discuss the upcoming Solvency II regulations for the European insurance industry. Again, we focus on their implications for the use of robust quantitative methods in financial risk management. Next, we consider the ingredients that we need for a robust quantitative risk management process. The first element are probability distances. We discuss definitions, properties, and examples, the main one being the Wasserstein metric. Probability distances are a prerequisite for obtaining many of the results in robust statistics. Before applying the robustness results, we discuss axiomatic approaches for risk measures, on probability spaces as well as on data. Finally, we combine all ingredients into the risk management procedure. Additionally, we discuss several—in particular simulation-based—approaches for the computation of the Solvency II capital requirement and set up a mathematical framework for the introduction of a new algorithm. We conduct an empirical study of its performance.