Sixty years ago, Landau discovered a paradoxical collisionless relaxation effect in plasmas. The Landau damping is now one of the cornerstones of classical plasma physics. From the mathematical point of view, it has remained elusive so far, since the best available results prove the existence of some damped solutions, without saying anything about their genericity. I shall report on new advances, and a whole new mathematical theory, for this problem. I will discuss the physical implications of these results. This is joint work with Clement Mouhot.Note: This is part of a series of related talks, there is additionally a colloquium and a math physics seminar.