  Recently Wolff obtained a sharp $L^2$ bilinear restriction theorem for
bounded subsets of the cone in general dimension. Here we adapt the argument of
Wolff to also handle subsets of ``elliptic surfaces'' such as paraboloids and
spheres. Except for an endpoint, this answers a conjecture of Machedon and
Klainerman, and also improves upon the known restriction theory for the
paraboloid and sphere.
