Uncertainty is ubiquitous in many complex systems. It is common to categorize uncertainty in aleatory and epistemic uncertainty, i.e., inherent variability and lack of knowledge, respectively. We discuss an example of each in the context of modelling and quantifying the uncertainties in PDE systems. In the first model, we discuss micro-scale variability of relaxation times in dispersive materials, and consider the effect of uncertainty on numerical analysis of discretization methods for Maxwell's equations, as well as on parameter estimation. In the second case, we describe a multi-reservoir control problem governed by the Saint-Venant equations, given limited inflow forecasts. We use uncertainty quantification techniques to inform a robust and reliable design optimization problem.