This dissertation is to address product design optimization including reliability-based design optimization (RBDO) and robust design with epistemic uncertainty. It is divided into four major components as outlined below. Firstly, a comprehensive study of uncertainties is performed, in which sources of uncertainty are listed, categorized and the impacts are discussed. Epistemic uncertainty is of interest, which is due to lack of knowledge and can be reduced by taking more observations. In particular, the strategies to address epistemic uncertainties due to implicit constraint function are discussed. Secondly, a sequential sampling strategy to improve RBDO under implicit constraint function is developed. In modern engineering design, an RBDO task is often performed by a computer simulation program, which can be treated as a black box, as its analytical function is implicit. An efficient sampling strategy on learning the probabilistic constraint function under the design optimization framework is presented. The method is a sequential experimentation around the approximate most probable point (MPP) at each step of optimization process. It is compared with the methods of MPP-based sampling, lifted surrogate function, and non-sequential random sampling. Thirdly, a particle splitting-based reliability analysis approach is developed in design optimization. In reliability analysis, traditional simulation methods such as Monte Carlo simulation may provide accurate results, but are often accompanied with high computational cost. To increase the efficiency, particle splitting is integrated into RBDO. It is an improvement of subset simulation with multiple particles to enhance the diversity and stability of simulation samples. This method is further extended to address problems with multiple probabilistic constraints and compared with the MPP-based methods. Finally, a reliability-based robust design optimization (RBRDO) framework is provided to integrate the consideration of design reliability and design robustness simultaneously. The quality loss objective in robust design, considered together with the production cost in RBDO, are used formulate a multi-objective optimization problem. With the epistemic uncertainty from implicit performance function, the sequential sampling strategy is extended to RBRDO, and a combined metamodel is proposed to tackle both controllable variables and uncontrollable variables. The solution is a Pareto frontier, compared with a single optimal solution in RBDO.