A multiobjective, risk-aware framework is developed for optimal path planning of autonomous underwater vehicles operating in uncertain current. The uncertainty in the current is described in terms of a finite ensemble of flow realizations. A new optimization framework is proposed that accounts for the full variability of the ensemble in a single optimization problem whose solution may not necessarily coincide with any of the optimal deterministic paths corresponding to individual ensemble members. We formulate stochastic problems that aim to minimize a risk measure of the travel time or energy consumption, using a flexible methodology that enables the user to seamlessly explore various objectives, ranging from risk neutral to risk averse. We illustrate the application of the proposed approach using two case studies based on synthetic 2-D settings, and one case involving a real-world problem in the Gulf of Aden. The results are analyzed to assess the value of stochastic solution, guide the selection of suitable risk measures, and demonstrate the impact of the risk measures on the resulting path and on the distribution of travel times.