Uncertainty is prevalent in engineering design, data-driven problems, and decision making broadly. Due to inherent risk-averseness and ambiguity about assumptions, it is common to address uncertainty by formulating and solving pessimistic optimization problems expressed using measures of risk and related concepts. Solidly rooted in convex analysis, risk measures furnish a general framework for handling uncertainty with significant computational and theoretical advantages. However, they are not always suitable and in fact can be counterproductive. An alternative, optimistic approach based on Rockafellian relaxations might be more suitable when data underpinning distributional assumptions is corrupted. We contrast the two perspectives, provide guidance about when one is preferrable to the other, and give several numerical examples including from computer vision and PDE-constrained optimization under uncertainty.