  We consider ergodic families of Schr\"odinger operators over base dynamics
given by strictly ergodic subshifts on finite alphabets. It is expected that
the majority of these operators have purely singular continuous spectrum
supported on a Cantor set of zero Lebesgue measure. These properties have
indeed been established for large classes of operators of this type over the
course of the last twenty years. We review the mechanisms leading to these
results and briefly discuss analogues for CMV matrices.
