Possibly the most common problem in statistics: How do we estimate the mean of a distribution given samples?
Light-tailed settings
A common approach is to invert concentration inequalities for the sample mean, see eg bounded scalar concentration and light-tailed, unbounded scalar concentration. In light-tailed settings, concentration is often synonymous with studying mean estimation, since the sample mean is a good estimator of the mean, and we are usually interested in concentration about the mean.
A separate approach is one based on hypothesis testing and game-theoretic statistics. See estimating means by betting. These are tighter than CIs based on Hoeffding and Bernstein inequalities. They are not in closed-form however, and require computational methods to estimate.
Heavy-tailed settings
The sample mean fails to be a good estimator in heavy-tailed settings (see Catoni’s 2012 paper) so one uses other estimators. For specific discussions of mean estimation in heavy-tailed settings, see scalar heavy-tailed mean estimation and multivariate heavy-tailed mean estimation.