  We study the region of complete localization in a class of random operators
which includes random Schr\"odinger operators with Anderson-type potentials and
classical wave operators in random media, as well as the Anderson tight-binding
model. We establish new characterizations or criteria for this region of
complete localization, given either by the decay of eigenfunction correlations
or by the decay of Fermi projections. (These are necessary and sufficient
conditions for the random operator to exhibit complete localization in this
energy region.) Using the first type of characterization we prove that in the
region of complete localization the random operator has eigenvalues with finite
multiplicity.
