This thesis develops and explores the connections between risk theory and robust optimization. Specifically, we show that there is a one-to-one correspondence between a class of risk measures known as coherent risk measures and uncertainty sets in robust optimization. An important consequence of this is that one may construct uncertainty sets, which are the critical primitives of robust optimization, using decision-maker risk preferences. In addition, we show some results on the geometry of such uncertainty sets. We also consider a more general class of risk measures known as convex risk measures, and show that these risk measures lead to a more flexible approach to robust optimization. In particular, these models allow one to specify not only the values of the uncertain parameters for which feasibility should be ensured, but also the degree of feasibility. We show that traditional, robust optimization models are a special case of this framework. As a result, this framework implies a family of probability guarantees on infeasibility at different levels, as opposed to standard, robust approaches which generally imply a single guarantee.