<p>This paper studies distributionally robust optimization (DRO) problems with polynomial uncertainty where the stochastic functions are polynomials of the random variables and the ambiguity set is defined as a ball in the space of probability measures centered at the empirical probability measure. This ball is measured using the optimal transport discrepancy, which includes Wasserstein’s distances as particular cases. The DRO problem can be equivalently reformulated as a linear conic optimization problem with nonnegative polynomial cones when the objective function is affine in the decision variables. We propose a Moment-SOS hierarchy relaxations method for solving the transformed problem and prove its convergent properties. Moreover, we can also obtain the worst-case probability measure. Numerical experiments are presented to illustrate the efficiency of our proposed algorithm.</p>