Optimization and uncertainty analyses used in conjunction with complex simulation models are important for using models to make predictions based on observations and for finding optimal designs or policies. Often these models can generate objective function surfaces with multiple local minima. Global Optimization and uncertainty analysis typically require a very large number of simulations, often thousands or tens of thousands. However, this number of simulations is not feasible for computationally expensive nonlinear simulation models. Our approach is to iteratively approximate the objective function or likelihood function f(x) with Radial Basis Functions (RBF) or other surrogate response surfaces during the search process. Our methods are derivative-free and can find local and global minima. It is this use of previously evaluated points f(xi) that is responsible for great savings in computational time. I will give results that compare these algorithms , including applications to complex simulations for groundwater remediation and carbon sequestration and for uncertainty quantification.