  We study the Ginzburg-Landau model of superconductivity in three dimensions
and for strong external magnetic fields. For magnetic field strengths above the
phenomenologically defined second critical field it is known from Physics that
superconductivity should be essentially restricted to a region near the
boundary. We prove that the expected region does indeed carry
superconductivity. Furthermore, we give precise energy estimates valid also in
the regime around the second critical field which display the transition from
bulk superconductivity to surface superconductivity.
