This talk will focus on uncertainty quantification in computational models, and specifically on the range of challenges in this field. These challenges include, to begin with, characterization of uncertain inputs given disparate and imperfect sources of information. They also include efficient sparse representation of random variables in a high-dimensional setting. Moreover, high dimensionality is a significant challenge for forward propagation of uncertainty, global sensitivity analysis, and surrogate construction over ranges of parameter variability, in complex computational models. Another key challenge, in particular for smooth polynomial representaitons of random variables, is the presence of discontinuous/bifurcative model behavior over parametric space. Finally, significant challenges are also present in the representation of uncertain quantities in dynamical systems with oscillatory behavior over long time horizons. I will range over these challenges, outlining our work in various elements of the picture, and describing promising research directions.