  We show that the nonlinear optical response reflects sensitively the
electronic structure of transition-metal surfaces and interfaces. $d$ and $s$
electrons may contribute rather differently to the second harmonic generation
(SHG) signal. This results from the different sensitivity of $d$ and $s$
electrons to surface and symmetry changes. Consequently, SHG for noble metals
shows a by far larger dependence on the polarization of the incoming light than
for transition metals like Fe, Co, Ni, in particular at lower frequencies. The
theoretical results are compared with recent measurements. We conclude that the
SHG yield is in addition to the nonlinear magneto-optical Kerr-effect a
sensitive fingerprint of the electronic structure at surfaces and interfaces.
