The idea of Stepwise Uncertainty Reduction (SUR) has appeared under various names and in various fields (psychophysics, computer vision, machine learning. . . ) throughout the eighties and the nineties. More recently, starting with the work of E. Vazquez and co-authors, it has been successfully applied to the sequential design of numerical experiments, in particular optimization and reliability analysis, based on Gaussian process priors. In a nutshell, a SUR sequential design greedily minimizes the expected value of some “measure of uncertainty” (e.g., the entropy or variance of some quantity of interest) in order to make it go to zero, hopefully as fast as possible. This talk will present recent results about the almost sure consistency of some SUR sequential designs, in particular under Gaussian process priors, and discuss the properties of uncertainty functionals (i.e., the functionals used to compute quantitative measures of uncertainty from posterior distributions) that make such results possible.