  We study Cox rings of K3-surfaces. A first result is that a K3-surface has a
finitely generated Cox ring if and only if its effective cone is polyhedral.
Moreover, we investigate degrees of generators and relations for Cox rings of
K3-surfaces of Picard number two, and explicitly compute the Cox rings of
generic K3-surfaces with a non-symplectic involution that have Picard number 2
to 5 or occur as double covers of del Pezzo surfaces.
