The universal Severi varieties of K3 surfaces parametrize nodal irreducible curves, of fixed geometric genus and degree, living on some K3 surface. I will first explain why it is expected that these varieties are irreducible, and why such a conjecture implies that a general algebraic K3 surface does not carry any rational endomorphism with degree >1. I will then give some irreducibility results for universal Severi varieties of hyperplane sections in the case of low genus K3 surfaces. Eventually, I will give a precise description of the latter varieties for genus 3 K3 surfaces (ie quartic hypersurfaces in P^3), from which I will derive some enumerative results.