This chapter describes the application of a computationally efficient uncertainty quantification approach, non-intrusive polynomial chaos (NIPC)-based stochastic expansions, for robust design under mixed (aleatory and epistemic) uncertainties and demonstrates this technique on robust design of a beam and on robust aerodynamic optimization. The approach utilizes stochastic response surfaces obtained with NIPC methods to approximate the objective function and the constraints in the optimization formulation. The objective function includes the stochastic measures, which are minimized simultaneously to ensure the robustness of the final design to both aleatory and epistemic uncertainties. The results of the optimization case studies show the computational efficiency and accuracy of the robust design with stochastic expansions, which may be applied to any stochastic optimization problem in science and engineering.