  We prove that if $X$ is a complex projective K3 surface and $g>0$, then there
exist infinitely many families of curves of geometric genus $g$ on $X$ with
maximal, i.e., $g$-dimensional, variation in moduli. In particular every K3
surface contains a curve of geometric genus 1 which moves in a non-isotrivial
family. This implies a conjecture of Huybrechts on constant cycle curves and
gives an algebro-geometric proof of a theorem of Kobayashi that a K3 surface
has no global symmetric differential forms.
