  Given a variety over a number field, are its rational points potentially
dense, i.e., does there exist a finite extension over which rational points are
Zariski dense? We study the question of potential density for symmetric
products of surfaces. Contrary to the situation for curves, rational points are
not necessarily potentially dense on a sufficiently high symmetric product. Our
main result is that rational points are potentially dense for the Nth symmetric
product of a K3 surface, where N is explicitly determined by the geometry of
the surface. The basic construction is that for some N, the Nth symmetric power
of a K3 surface is birational to an abelian fibration over P^N. It is an
interesting geometric problem to find the smallest N with this property.
