Global sensitivity analysis (GSA) offers a flexible framework for understanding the structural importance of uncertain parameters in mathematical models. This dissertation focuses on forward and inverse problems arising in uncertainty quantification and the computation of Sobol’ indices, measures of variance-based sensitivity. The models involved in these prob- lems are often computationally expensive to evaluate. Sensitivity analysis then comes at an unreasonable cost. A preferred workaround is to create a surrogate of the model that is less cumbersome to work with. Surrogate methods that accelerate GSA are proposed and studied in this dissertation. We consider connections between function approximation and the functional analysis of variance (ANOVA) decomposition. We provide insight into the challenge of efficiently uncovering model structure through interaction screening and distance to additivity. A new class of surrogate models is introduced, using random weight neural networks for surrogate-assisted GSA, presenting analytical formulas for Sobol’ indices. The proposed algorithm enhances accuracy through weight sparsity selection, as shown by its application to forward problems derived from ordinary differential equation systems. The dissertation tackles sensitivity analysis in Bayesian inverse problems, a challenging domain due to computational and methodological complexities. A framework for variance-based sensitivity analysis of Bayesian inverse problems with respect to prior hyperparameters is introduced, along with an efficient algorithm combining importance sampling and surrogate modeling. The approach is demonstrated on a nonlinear Bayesian inverse problem in epidemiology, showcasing its effectiveness in quantifying uncertainty in posterior statistics.