  The article reviews recent developments in the theory of fluctuations and
correlations of energy levels and eigenfunction amplitudes in diffusive
mesoscopic samples. Various spatial geometries are considered, with emphasis on
low-dimensional (quasi-1D and 2D) systems. Calculations are based on the
supermatrix sigma-model approach. The method reproduces, in so-called zero-mode
approximation, the universal random matrix theory (RMT) results for the
energy-level and eigenfunction fluctuations. Going beyond this approximation
allows us to study system-specific deviations from universality, which are
determined by the diffusive classical dynamics in the system. These deviations
are especially strong in the far ``tails'' of the distribution function of the
eigenfunction amplitudes (as well as of some related quantities, such as local
density of states, relaxation time, etc.). These asymptotic ``tails'' are
governed by anomalously localized states which are formed in rare realizations
of the random potential. The deviations of the level and eigenfunction
statistics from their RMT form strengthen with increasing disorder and become
especially pronounced at the Anderson metal-insulator transition. In this
regime, the wave functions are multifractal, while the level statistics
acquires a scale-independent form with distinct critical features. Fluctuations
of the conductance and of the local intensity of a classical wave radiated by a
point-like source in the quasi-1D geometry are also studied within the
sigma-model approach. For a ballistic system with rough surface an
appropriately modified (``ballistic'') sigma-model is used. Finally, the
interplay of the fluctuations and the electron-electron interaction in small
samples is discussed, with application to the Coulomb blockade spectra.

Disordered Systems and Neural Networks;Mesoscale and Nanoscale Physics