Abstract: Scientific inference often involves inferring internal key parameters that determine the outcome of a complex physical phenomenon. The data themselves may come in the form of a labeled set that implicitly encodes the likelihood function; for example, in the form of (i) pairs of parameters and observable data according to a mechanistic (simulator) model, or as (ii) observed data and parameters 'measured' with high precision via an auxiliary experiment. We refer to inference in both intractable likelihood settings as "Likelihood-Free Inference'" (LFI). The application of neural density estimators and generative models to scientific LFI settings is becoming increasingly widespread. However, high-posterior density (HPD) regions derived from these density estimators do not necessarily have a high probability of including the true parameter of interest, even if the posterior is well-estimated and the labeled data have the same distribution as the target distribution. Furthermore, if the prior distribution is poorly specified, then the HPD regions could severely undercover and/or be biased, thereby leading to misleading scientific conclusions. In this talk, I will present new LFI methodology and algorithms for leveraging neural density estimators to produce confidence regions that have (i) nominal frequentist coverage for any value of the (unknown) parameter, even with just one observation (sample size n=1), and (ii) smaller average area (yielding higher constraining power) if the prior is well-specified. I will illustrate our methods on examples from astronomy and high-energy physics, and discuss where we stand and what challenges still remain.