In uncertainty quantification (UQ), physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution. In forward UQ, one aims to understand how uncertainty in model inputs affects uncertainty in model solutions. Sampling methods require the repeated numerical solution of the PDE for different samples of the inputs, so when the cost of solving the problem for just one sample is already expensive, obtaining accurate uncertainty assessments becomes infeasible. Accurate yet cheap(er) surrogate models are required. In this talk, I will give a brief survey of some of the key advances in the numerical analysis community over the last two decades that allowed us to continue to push the envelope in terms of the dimensionality and complexity of the problems that can be handled on standard computers with modest resources. These include multilevel techniques, sparse grids and adaptivity. As advances in the field of Operator Learning, and Deep Learning more generally, look to be fast eclipsing some of these past achievements, we end with a brief discussion about the future role of numerical analysis in the design of numerical solution algorithms for physics-based problems.