We examine characterizations of Pareto-optimal risk sharing when agents' preferences are risk averse. Specifically, we consider two broad classes of preferences, both of which admit the convex distortion risk measure as a special case. By introducing robustness in terms of the distortion function, we recover the class of coherent risk measures. While an explicit characterization of Pareto optimality is difficult to obtain, we present an implicit result, as well as an efficient algorithm to numerically solve for optimal risk-sharing allocations. Our numerical results suggest that the structure of optimal allocations can be quite complicated, even in relatively simple scenarios. On the other hand, introducing a utility function to the distortion risk measure gives the class of rank-dependent utilities, which is well-studied in the decision theory literature. In this setting, we first present a result that links Pareto-optimality of allocations to the Expected Shortfall, and we obtain the known results on expected utilities and dual utilities as corollaries. When certain monotonicity conditions are satisfied, we also provide a closed-form expression for Pareto optimality, and an algorithm to efficiently calculate any allocation on the Pareto frontier. We apply this process in a numerical illustration to show how Pareto improvements can be found over more simplistic allocation mechanisms such as the 1/n risk-sharing rule.