In the design and analysis of computers experiments, "inversion" refers, broadly speaking, to the problem of finding the values of the inputs of a model that lead to outputs with given properties. In such problems, the object of interest is a subset of the input space, or its volume under a given measure. Robust formulations of the inversion problem, where the objective is to solve an inversion problem with respect to a given subset of the input variables in the presence of uncertainty on the others, are particularly important in many applications. We focus in this talk on a particular robust formulation, which we call "Reliability-Based Inversion" (RBI), in which the objective is to solve an inversion problem for a quantile of the output with respect to the uncertain variables of the model (or, equivalently, for the probability of exceeding a given threshold). In this setting, we use the Stepwise Uncertainty Reduction (SUR) paradigm, which has proved fruitful for simpler variants of the inversion problem, to construct efficient sequential sampling strategies for the RBI problem.