Robust optimization is a set-based approach to handling uncertainty within optimization problems. The focus of this talk is to unify the robust optimization approach with risk-theoretic concepts. In the first part of the talk, we propose a methodology based on decision-maker risk preferences for constructing uncertainty sets within the framework of robust optimization. We show that there is a bijection between uncertainty sets and a particular class of risk measures. In the second part of the talk, we develop a flexible approach to robust optimization based on a more general class of risk measures. This approach allows one to specify not only the values of the uncertain parameters for which feasibility should be ensured, but also the *degree* of feasibility. Finally, we demonstrate the efficacy of this approach on a real-world portfolio optimization problem.