  In connection with the restriction problem in $\mathbb R^n$ for hypersurfaces
including the sphere and paraboloid, the bilinear (adjoint) restriction
estimates have been extensively studied. However, not much is known about such
estimates for surfaces with codimension (and dimension) larger than one. In
this paper we show sharp bilinear $L^2 \times L^2 \to L^q$ restriction
estimates for general surfaces of higher codimension. In some special cases, we
can apply these results to obtain the corresponding linear estimates.
