Birational geometry of hypersurfaces in products of projective spaces

We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in , we describe their nef, movable and effective cones and determine when they are Mori dream spaces. Using this, 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.