This talk has as main purpose the estimation of the mean of heavy-tailed distributions. This field is not new and it is known from Huber’s work that it is possible to create estimators insensible to outliers and that satisfy a Central Limit Theorem. Recently, new robust estimators have been proposed that satisfy concentration inequalities – hence at finite and fixed $n$ – with sub-Gaussian speed under a small moment assumption (finite variance). The robust estimation of the mean for real valued random variables is a starting point for extensions of robust estimators to more complex probability fields. The definition of a robust estimator and its concentration in the case of multivariate (in $R^d$) distributions will be the main purpose of this talk.