Traditional algorithms for optimal control of nonlinear systems under significant state or parameter uncertainty typically require that the probability density of the state be evolved through time using Monte Carlo methods or explicit techniques such as the Frobenius-Perron operator. Such methods can be computationally intensive for systems with a large number of uncertain parameters or initial states. This talk presents an alternative approach for solving optimal control problems under state or parameter uncertainty using the Koopman operator. In the proposed approach, rather than evolving the state density forward and computing expected values with cost and constraint functions, the cost and constraints are instead pulled back to the initial time through the action of the Koopman operator. The expected values can then be computed with respect to the initial state probability density. This technique has several numerical advantages over Monte Carlo and other explicit approaches. Following an overview of the Koopman operator method, the talk will illustrate application of the algorithm through several examples.  These include toy problems such as optimizing the path of a bouncing ball and path planning for a Dubins car. More complex examples will also be presented in which the Koopman operator approach is applied to airdrop mission planning as well as booster ignition timing optimization for a multi-stage rocket. Overall, these examples highlight the real-world applicability of the Koopman operator approach as well as its advantages over alternative methods.