Computational uncertainty quantification (UQ) is regarded as an integral part of predictive science, an emerging discipline concerned with assessing the predictability of mathematical and computational tools in the presence of uncertainty. In this talk, I will present a UQ methodology embedded in a new hybrid fuzzy-stochastic framework for the prediction of physical events described by partial differential equations (PDEs) and subject to a mixture of aleatoric and epistemic uncertainty. In the new framework, uncertain input parameters are characterized by random fields with fuzzy moments. This will result in a new class of PDEs with hybrid fuzzy-stochastic parameters, coined fuzzy-stochastic PDEs, for which forward and inverse problems need to be solved. I will demonstrate the importance and feasibility of the new methodology by applying it to a complex problem: prediction of the response of materials with hierarchical microstructure to external forces.