  We discuss the role of rare fluctuation effects in quantum condensed matter
systems. In particular, we present recent numerical results of the effect of
resonant states in Anderson's original model of electron localization. We find
that such resonances give rise to anomalous behavior of eigenstates not just
far in the Lifshitz tail, but rather for a substantial fraction of eigenstates,
especially for intermediate disorder. The anomalous behavior includes
non-analyticity in various properties as a characteristic. The effect of
dimensionality on the singularity, which is present in all dimensions, is
described, and the behavior for bounded and unbounded disorder is contrasted.
