  We study the birational properties of hypersurfaces in products of projective
spaces. In the case of hypersurfaces in P^m x P^n, we describe their nef,
movable and effective cones and determine when they are Mori dream spaces.
Using these results, we give new simple examples of non-Mori dream spaces and
analogues of Mumford's example of a strictly nef line bundle which is not
ample.
