  We present a theory of the quasi two-dimensional electron gas (2DEG) systems
which appear near the surface of SrTiO$_3$ when a large external electric field
attracts carriers to the surface. We find that non-linear and non-local
screening by the strongly polarizable SrTiO$_3$ lattice plays an essential role
in determining 2DEG properties. The electronic structure always includes weakly
bound bulk-like bands that extend over many SrTiO$_3$ layers. At 2D
carrier-densities exceeding $\sim 10^{14} {\rm cm}^{-2}$ tightly bound bands
emerge that are confined within a few layers of the surface.
