  In general, not much is known about the arithmetic of K3 surfaces. Once the
geometric Picard number, which is the rank of the Neron-Severi group over an
algebraic closure of the base field, is high enough, more structure is known
and more can be said. However, until recently not a single K3 surface was known
to have geometric Picard number one. We give explicit examples of such surfaces
over the rational numbers. This solves an old problem that has been attributed
to Mumford. The examples we give also contain infinitely many rational points,
thereby answering a question of Swinnerton-Dyer and Poonen.
