K3 surfaces are two dimensional Calabi-Yau manifolds. Their moduli space is of interest in algebraic geometry, but also has connections with number theory and string theory. I will discuss ongoing joint work with Alina Marian and Rahul Pandharipande aimed at studying the tautological ring of the moduli space of K3 surfaces. In particular, I will discuss different notions of tautological classes. Next, I will explain a method of deriving relations between tautological classes via the geometry of the relative Quot scheme.