"In joint work with Chen, we proved that on any K3 surface one can produce curves of any fixed geometric genus g, each of which deforms maximally in moduli, i.e. in a g-dimensional family of M_g. In this talk I will discuss this and some related results, and various applications, in particular to the existence of symmetric differentials on K3s. The key inputs in the proof are the existence of infinitely many rational curves on a K3 (recently obtained the remaining cases jointly with Chen-Liedtke) and the logarithmic Bogomolov-Miyaoka-Yau inequality which provides some (very weak) control of the singularities of these rational curves. "