The main contributions of the present thesis are novel computational methods related to uncertainty quantification, inverse problems and reduced order modeling in engineering. In the first chapter, we describe a framework to optimize an engineering system under large uncertainties. The optimization problem being recast as a sampling problem, the use of advanced sampling schemes associated with a hierarchical approach using approximate models enables an efficient identification of design values; along with corresponding sensitivity and robustness information. The second chapter deals with the solution of Bayesian inverse problems, in which unknown parameter values in a model are being inferred from uncertain measurements of the output of the system of interest. A reduced order model interpolation scheme, based on differential geometric ideas, enables faster computations during the posterior sampling process while maintaining a high accuracy. Finally, the last chapter proposes a solution to the snapshot selection problem in reduced order modeling, namely how to select the parameters that represent best the system of interest in the parameter space. The approach chosen is to interpret the parameter space as a Riemannian manifold, with a sensitivity related metric emphasizing regions with more information. The numerical applications chosen for each of those problems are engineering oriented, with the corresponding models being discretized using the finite element method.