In this talk we first review the framework of robust estimation where robustness can be with respect to outliers or heavy-tailed noise. Then, we consider the sparse linear model where some of the observations can be corrupted and the noise heavy-tailed. After deriving the minimax quadratic risk for estimation of the signal, we propose a practical and fully adaptive procedure that is optimal. Our procedure corresponds to solving a novel penalized pivotal estimation problem. As a result, we develop a new method that is not only minimax optimal and robust but also enjoys sub-Gaussian deviations even in the presence of heavy-tailed noise. Joint with S. Minsker and L. Wang.