  Neural point estimators are neural networks that map data to parameter point
estimates. They are fast, likelihood free and, due to their amortised nature,
amenable to fast bootstrap-based uncertainty quantification. In this paper, we
aim to increase the awareness of statisticians to this relatively new
inferential tool, and to facilitate its adoption by providing user-friendly
open-source software. We also give attention to the ubiquitous problem of
making inference from replicated data, which we address in the neural setting
using permutation-invariant neural networks. Through extensive simulation
studies we show that these neural point estimators can quickly and optimally
(in a Bayes sense) estimate parameters in weakly-identified and
highly-parameterised models with relative ease. We demonstrate their
applicability through an analysis of extreme sea-surface temperature in the Red
Sea where, after training, we obtain parameter estimates and bootstrap-based
confidence intervals from hundreds of spatial fields in a fraction of a second.
