  A technically convenient signature of localization, exhibited by discrete
operators with random potentials, is exponential decay of the fractional
moments of the Green function within the appropriate energy ranges. Known
implications include: spectral localization, absence of level repulsion, strong
form of dynamical localization, and a related condition which plays a
significant role in the quantization of the Hall conductance in two-dimensional
Fermi gases. We present a family of finite-volume criteria which, under some
mild restrictions on the distribution of the potential, cover the regime where
the fractional moment decay condition holds. The constructive criteria permit
to establish this condition at spectral band edges, provided there are
sufficient `Lifshitz tail estimates' on the density of states. They are also
used here to conclude that the fractional moment condition, and thus the other
manifestations of localization, are valid throughout the regime covered by the
``multiscale analysis''. In the converse direction, the analysis rules out fast
power-law decay of the Green functions at mobility edges.
