Although a variety of uncertainty propagation methods exist for estimating the statistical moments and the probability of failure in design under uncertainty, current methods suffer from their limitations in providing accurate and efficient solutions to high-dimension problems with interactions of random variables. A new sparse grid based uncertainty propagation method is proposed in this work to overcome this difficulty. The existing sparse grid technique, originally invented for numerical integration and interpolation, is extended to uncertainty propagation in the probabilistic domain. In particular, the concept of Sparse Grid Numerical Integration (SGNI) is extended for estimating the first two moments of performance in robust design, while the Sparse Grid Interpolation (SGI) is employed to determine failure probability by interpolating the limit-state function at the Most Probable Point (MPP) in reliability analysis. The proposed methods are demonstrated by several high-dimension mathematical examples with notable variate interactions and one complex multidisciplinary rocket design problem. Results show that the use of sparse grid methods works better than popular counterparts. Furthermore, the automatic sampling, special interpolation process, and dimension-adaptivity feature make SGI more flexible and efficient than using the uniform sample based metamodeling techniques.