This talk will be about dynamical properties of the Teichmüller geodesic flow and the SL(2,R) action on moduli spaces of Abelian differentials. Back in 2006, Avila, Gouëzel and Yoccoz proved that the geodesic flow is exponentially mixing. Here we are interested in not just mixing, but exponential multiple mixing. After introducing the moduli spaces, the group action on them and notions of mixing, I will go through the idea of the proof for exponential multiple mixing. If time permits, I will also talk about an application to a central limit theorem.