Integrated Computational Materials Engineering (ICME) hinges on engineering microstructural features into the materials design process where the overarching goal is to search for materials with superior structure-level performance while systematically accounting for various sources of uncertainties, including those introduced by manufacturing processes and simulation models. Quantification of manufacturing induced uncertainties is significantly challenging since they are multi-dimensional, spread across different length-scales, spatially correlated, and embody different characteristics (e.g., topological vs. property-related). In this talk, we address these challenges by presenting a non-intrusive computational approach for multiscale and multidimensional uncertainty quantification. We introduce the top-down sampling method that allows to model non-stationary and continuous (but not differentiable) spatial variations of uncertainty sources by creating nested random fields. We employ multi-response Gaussian random processes in top-down sampling and leverage sensitivity analyses and supervised learning to address the considerable computational costs of multiscale simulations. To quantify the model uncertainty, a physics-informed modular Bayesian approach is employed where the lack of experimental and simulation resources is addressed by enforcing certain physical constraints on the functional forms of the simulator and its potential discrepancies in the Bayesian analyses. Examples with carbon fiber reinforced polymer (CFRP) composites will be used to illustrate the broader impact of the uncertainty quantification methods in multiscale materials systems.