  The physics of quantum degenerate Fermi gases in uniform as well as in
harmonically trapped configurations is reviewed from a theoretical perspective.
Emphasis is given to the effect of interactions which play a crucial role,
bringing the gas into a superfluid phase at low temperature. In these dilute
systems interactions are characterized by a single parameter, the s-wave
scattering length, whose value can be tuned using an external magnetic field
near a Feshbach resonance. The BCS limit of ordinary Fermi superfluidity, the
Bose-Einstein condensation (BEC) of dimers and the unitary limit of large
scattering length are important regimes exhibited by interacting Fermi gases.
In particular the BEC and the unitary regimes are characterized by a high value
of the superfluid critical temperature, of the order of the Fermi temperature.
Different physical properties are discussed, including the density profiles and
the energy of the ground-state configurations, the momentum distribution, the
fraction of condensed pairs, collective oscillations and pair breaking effects,
the expansion of the gas, the main thermodynamic properties, the behavior in
the presence of optical lattices and the signatures of superfluidity, such as
the existence of quantized vortices, the quenching of the moment of inertia and
the consequences of spin polarization. Various theoretical approaches are
considered, ranging from the mean-field description of the BCS-BEC crossover to
non-perturbative methods based on quantum Monte Carlo techniques. A major goal
of the review is to compare the theoretical predictions with the available
experimental results.
