  We investigate the universal Severi variety of rational curves on K3
surfaces, which parametrises irreducible rational curves in a fixed class on
varying K3 surfaces of fixed genus. We investigate the conjecuted
irreducibility of this space in the case of primitive classes using the
deformation theory of stable maps and obtain partial results. In an appendix we
give a short proof of the irreducibility of universal Severi varieties of high
genus curves on K3 surfaces.
