Given n independent, identically distributed copies of a random variable, one is interested in estimating the expected value. Perhaps surprisingly, there are still open questions concerning this very basic problem in statistics.

In this talk we are primarily interested in non-asymptotic sub-Gaussian estimates for potentially heavy-tailed random variables. We discuss various estimates and extensions to high dimensions, empirical risk minimization, and multivariate problems. This talk is based on joint work with Emilien Joly, Luc Devroye, Matthieu Lerasle, and Roberto Imbuzeiro Oliveira.