This thesis focuses on the problem of evaluating high-dimensional integrals arising in optimization under uncertainty. Uncertainties in the input data affect the behavior of the physical system and need to be accounted for at the design stage or in the way the system is controlled. This translates into evaluating integrals of the quantities of interest with respect to the random parameters. This task becomes challenging when the dimension of the random parameters is high. Without guidelines for the choice of favorable integration methods the optimization algorithm might encounter prohibitively high computational cost. This thesis provides a comprehensive overview of methods for high-dimensional integration and exposes their relative strengths and weaknesses. Emphasis is placed on problems with moderately high dimension and with non-smoothness. The performance of integration methods in high dimension is assessed on several simple model problems.