  This paper is a short review on the foundations and recent advances in the
microscopic Fermi-liquid (FL) theory. We demonstrate that this theory is built
on five identities, which follow from conservation of total charge (particle
number), spin, and momentum in a translationally and $SU(2)$-invariant FL.
These identities allows one to express the effective mass and quasiparticle
residue in terms of an exact vertex function and also impose constraints on the
"quasiparticle" and "incoherent" (or "low-energ" and "high-energy")
contributions to the observable quantities. Such constraints forbid certain
Pomeranchuk instabilities of a FL, e.g., towards phases with order parameters
that coincide with charge and spin currents. We provide diagrammatic
derivations of these constraints and of the general (Leggett) formula for the
susceptibility in arbitrary angular momentum channel, and illustrate the
general relations through simple examples treated in the perturbation theory.
