The Chow ring of hyper-Kähler varieties should enjoy similar properties as the Chow ring of abelian varieties. In particular, a Beauville type decomposition is believed (by Beauville himself) to exist for all hyper-Kähler varieties. In this talk, we discuss a general approach towards the Beauville type decomposition of the Chow ring. We carry it out explicitly for the Hilbert scheme of points of K3 surfaces, and prove the multiplicativity of the resulting decomposition. Joint work with Andrei Negut and Georg Oberdieck.