  A new approach to the single-band Hubbard model is described in the general
context of many-body theories. It is based on enforcing conservation laws, the
Pauli principle and a number of crucial sum-rules. More specifically, spin and
charge susceptibilities are expressed, in a conserving approximation, as a
function of two constant irreducible vertices whose values are found
self-consistently. The Mermin-Wagner theorem in two dimensions is automatically
satisfied. The effect of collective modes on single-particle properties is then
obtained by a paramagnon-like formula that is consistent with the two-particle
properties in the sense that the potential energy obtained from $Tr\Sigma G$ is
identical to that obtained using the fluctuation-dissipation theorem for
susceptibilities. The vertex corrections are included through constant
irreducible vertices. The theory is in quantitative agreement with Monte Carlo
simulations for both single-particle and two-particle properties. In the
two-dimensional renormalized classical regime, spin fluctuations lead to
precursors of antiferromagnetic bands (shadow bands) and to the destruction of
the Fermi-liquid quasiparticles in a wide temperature range above the
zero-temperature phase transition. The analogous phenomenon of pairing
pseudogap can occur in the attractive model in two dimensions when the pairing
fluctuations become critical. Other many-body approaches are critically
compared. It is argued that treating the spin fluctuations as if there was a
Migdal's theorem can lead to wrong predictions, in particular with regard to
the the single-particle pseudogap.
