Bilinear restriction estimates have 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
)
(L^p,L^p)
spherical restriction theorem of Wolff from
p
>
42
/
11
p > 42/11
to
p
>
34
/
9
p > 34/9
, and also obtain a sharp
(
L
p
,
L
q
)
(L^p,L^q)
spherical restriction theorem for
q
>
4
−
5
27
q> 4 - \frac {5}{27}
.