We discuss an interplay between the Fermi-liquid (FL) theory and diagrammatic perturbative approach to interacting Fermi systems. In the FL theory for Galilean-invariant systems, mass renormalization ${m}^{\ensuremath{\ast}}/m$ comes exclusively from fermions at the Fermi surface. We show that in a diagrammatic perturbation theory the same result for ${m}^{\ensuremath{\ast}}/m$ comes from fermions both at and away from the Fermi surface. The equivalence of the FL and perturbative approaches is based on a particular relation between self-energy contributions from high- and low-energy fermions. We argue that care has to be exercised in the renormalization-group approach to a FL in order not to miss the high-energy contribution to ${m}^{\ensuremath{\ast}}/m$. As particular examples, we discuss ${m}^{\ensuremath{\ast}}/m$ and the quasiparticle residue
Z for two-dimensional and three-dimensional systems with both SU(2) and
SU(N) symmetries, and with a short-range interaction. We derive an expression for the anisotropic part of the Fermi-liquid vertex in the large-
N limit of the
SU(N) case.
We examine a two-dimensional Fermi liquid with a Fermi surface which touches the Umklapp surface first at the four points as the electron density is increased. Umklapp processes at the four patches near lead the renormalization group equations to scale to strong coupling, resembling the behaviour of a two-leg ladder at half-filling. The incompressible character of the fixed point causes a breakdown of Landau theory at these patches. A further increase in density spreads the incompressible regions so that the open Fermi surface shrinks to four disconnected segments. This non-Landau state, in which parts of the Fermi surface are truncated to form an insulating spin liquid, has many features in common with phenomenological models recently proposed for the cuprate superconductors.