The aim of this minisymposium is to discuss recent progress in the analysis and application of high-order methods for approximating the parameter-to-solution map in the context of random and parametric PDEs; such problems arise e.g. in uncertainty quantification and PDE-based optimization and control. We welcome theoretical, methodological, and application-oriented contributions on, among others, polynomial/rational approximations, kernel and Gaussian process regression methods, low-rank, proper generalized decomposition and reduced order models, as well as scientific-machine-learning-based methods for operator learning.