When using computational models to design devices or systems, appropriate modeling of uncertainties is crucial to ensure robust and reliable designs. This talk offers a survey of uncertainty quantification and reliability analysis techniques and demonstrates how they can be tightly integrated with optimization algorithms to determine designs that meet reliability and robustness constraints in addition to optimality criteria. Such optimization under uncertainty (OUU) capabilities, developed and implemented in Sandia National Laboratories' DAKOTA toolkit, will be surveyed. Novel problem formulations for reliability-based design optimization in realistic physical applications will be presented and examples of how advanced reliability methods can provide more accurate estimates of output uncertainty given.