  Bilinear restriction estimates have been appeared in work of Bourgain,
Klainerman, and Machedon. In this paper we develop the theory of these
estimates (together with the analogues for Kakeya estimates). As a consequence
we improve the $(L^p,L^p)$ spherical restriction theorem of Wolff from $p >
42/11$ to $p > 34/9$, and also obtain a sharp $(L^p,L^q)$ spherical restriction
theorem for $q > 4 - 5/27$.
