Fermi liquid is a generic term for a quantum mechanical liquid of fermions that arises under certain physical conditions when the temperature is sufficiently low. The interaction between the particles of the many-body system does not need to be small (see e.g. electrons in a metal). The phenomenological theory of Fermi liquids, which was introduced by the Soviet physicist Lev Davidovich Landau in 1956, explains why some of the properties of an interacting fermion system are very similar to those of the Fermi gas (i.e. non-interacting fermions), and why other properties differ.
Important examples of where Fermi liquid theory has been successfully applied are most notably electrons in most metals and Liquid He-3 . Liquid He-3 is a Fermi liquid at low temperatures (but not low enough to be in its superfluid phase.) He-3 is an isotope of Helium, with 2 protons, 1 neutron and 2 electrons per atom; because there is an odd number of fermions inside the atom, the atom itself is also a fermion. The electrons in a normal (non-superconducting) metal also form a Fermi liquid.
Similarities to Fermi gas
The Fermi liquid is qualitatively analogous to the non-interacting Fermi gas, in the following sense: The system's dynamics and thermodynamics at low excitation energies and temperatures may be described by substituting the non-interacting fermions with so-called quasiparticles, each of which carries the same spin, charge and momentum as the original particles. Physically these may be thought of as being particles whose motion is disturbed by the surrounding particles and which themselves perturb the particles in their vicinity. Each many-particle excited state of the interacting system may be described by listing all occupied momentum states, just as in the non-interacting system. As a consequence, quantities such as the heat capacity of the Fermi liquid behave qualitatively in the same way as in the Fermi gas (e.g. the heat capacity rises linearly with temperature).
Differences from Fermi gas
The following differences to the non-interacting Fermi gas arise:
The energy of a many-particle state is not simply a sum of the single-particle energies of all occupied states. Instead, the change in energy for a given change δnk in occupation of states k contains terms both linear and quadratic in δnk (for the Fermi gas, it would only be linear, δnkεk, where εk denotes the single-particle energies). The linear contribution corresponds to renormalized single-particle energies, which involve, e.g., a change in the effective mass of particles. The quadratic terms correspond to a sort of "mean-field" interaction between quasiparticles, which is parameterized by so-called Landau Fermi liquid parameters and determines the behaviour of density oscillations (and spin-density oscillations) in the Fermi liquid. Still, these mean-field interactions do not lead to a scattering of quasi-particles with a transfer of particles between different momentum states.
Specific heat and compressibility
Specific heat, compressibility, spin-susceptibility and other quantities show the same qualitative behaviour (e.g. dependence on temperature) as in the Fermi gas, but the magnitude is (sometimes strongly) changed.
In addition to the mean-field interactions, some weak interactions between quasiparticles remain, which lead to scattering of quasiparticles off each other. Therefore, quasiparticles acquire a finite lifetime. However, at low enough energies above the Fermi surface, this lifetime becomes very long, such that the product of excitation energy (expressed in frequency) and lifetime is much larger than one. In this sense, the quasiparticle energy is still well-defined (in the opposite limit, Heisenberg's uncertainty relation would prevent an accurate definition of the energy).
The structure of the "bare" particle's (as opposed to quasiparticle) Green's function is similar to that in the Fermi gas (where, for a given momentum, the Green's function in frequency space is a delta peak at the respective single-particle energy). The delta peak in the density-of-states is broadened (with a width given by the quasiparticle lifetime). In addition (and in contrast to the quasiparticle Green's function), its weight (integral over frequency) is suppressed by a quasiparticle weight factor 0 < Z < 1. The remainder of the total weight is in a broad "incoherent background", corresponding to the strong effects of interactions on the fermions at short time-scales.
The distribution of particles (as opposed to quasiparticles) over momentum states at zero temperature still shows a discontinuous jump at the Fermi surface (as in the Fermi gas), but it does not drop from 1 to 0: the step is only of size Z.
In a metal the resistance at low temperatures is dominated by electron-electron scattering in combination with Umklapp scattering. For a Fermi liquid, the resistance from this mechanism varies as T2, which is often taken as an experimental check for Fermi liquid behaviour (in addition to the linear temperature-dependence of the specific heat), although it only arises in combination with the lattice.
Instabilities of the Fermi Liquid
The experimental observation of exotic phases in strongly correlated systems has triggered an enormous effort from the theoretical community to try to understand their microscopical origin. One possible route to detect instabilities of a FL is precisely the analysis done by Pomeranchuk. Due to that, the Pomeranchuk instability has been studied by several authors  with different techniques in the last few years and in particular, the instability of the FL towards the nematic phase was investigated for several models.
1. ^ H.J. Schulz, Fermi liquids and non-Fermi liquids, arXiv:cond-mat/9503150v2 (1995)
* Classical fluids