We present a reinforcement learning (RL) approach for robust optimization of risk-aware performance criteria. To allow agents to express a wide variety of risk-reward profiles, we assess the value of a policy using rank dependent expected utility (RDEU). RDEU allows agents to seek gains, while simultaneously protecting themselves against downside risk. To robustify optimal policies against model uncertainty, we assess a policy not by its distribution but rather by the worst possible distribution that lies within a Wasserstein ball around it. Thus, our problem formulation may be viewed as an actor/agent choosing a policy (the outer problem) and the adversary then acting to worsen the performance of that strategy (the inner problem). We develop explicit policy gradient formulae for the inner and outer problems and show their efficacy on three prototypical financial problems: robust portfolio allocation, benchmark optimization, and statistical arbitrage.