  We consider continuum random Schr\"odinger operators of the type $H_{\omega}
= -\Delta + V_0 + V_{\omega}$ with a deterministic background potential $V_0$.
We establish criteria for the absence of continuous and absolutely continuous
spectrum, respectively, outside the spectrum of $-\Delta +V_0$. The models we
treat include random surface potentials as well as sparse or slowly decaying
random potentials. In particular, we establish absence of absolutely continuous
surface spectrum for random potentials supported near a one-dimensional surface
(``random tube'') in arbitrary dimension.
