The prediction of model outcomes frequently hinges on understanding the variability introduced by input parameters to the model. Such parametric influences can encode stochastic fluctuations, model-form error, and geometric uncertainty. Therefore, developing robust techniques for predicting the uncertainty resulting from this parametric variation is of utmost importance. The goal of these lectures is to discuss the role that approximation theory plays in modern UQ strategies for prediction of parametric variability. We concentrate on polynomial representations, which frequently entail use of a high-order generalized Polynomial Chaos approximation. During the first lecture, our discussion will touch on methods that can be used to predict parametric variation: regression models, sparse representations and grid layouts, and intrusive versus non-intrusive approaches. The major challenge with all these approaches is that the number of parameters can be very large, and this quickly renders many direct methods computationally untenable. In the second lecture, we will present a more detailed analysis of the advantages and limitations of these methods, and briefly survey some state-of-the-art approaches leading to modern research avenues for UQ approximation methods.