Uncertainty is inevitable in computer-based simulations. To provide more reliable predictions for the behavior of complex systems or optimal designs for the large structures, understanding and quantifying the uncertainty in simulations is critical. In this talk, we will focus on two of the main aspects of uncertainty quantification (UQ): model form UQ (backward UQ or model calibration) and application of UQ to material design. Specifically, for model form UQ, observations are available and physical constraints are incorporated into model correction process to enforce the important physical properties of the underlying system. The estimation of both model output and model parameters can be improved. For the application of UQ, we propose a robust inverse design procedure for the optimal morphology of nanoparticles in Plasmonics. Specifically, we use a global sensitivity analysis method to identify the important random variables and consider the non-important ones as deterministic, and consequently reduce the dimension of the stochastic space. In addition, we apply the generalized polynomial chaos expansion method for constructing computationally cheaper surrogate models to approximate and replace the full simulations.