**Abstract** : 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 irreducible vertices whose values are found by imposing the local Pauli principle 〈n2↑〉= 〈n↑〉 as well as the local-moment sum-rule and consistency with the equations of motion in a local-field approximation. 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 ΣG is identical to that obtained using the fluctuation-dissipation theorem for susceptibilities. Since there is no Migdal theorem controlling the effect of spin and charge fluctuations on the self-energy, the required vertex corrections are included. It is shown that the theory is in quantitative agreement with Monte Carlo simulations for both single-particle and two-particle properties. The theory predicts a magnetic phase diagram where magnetic order persists away from half-filling but where ferromagnetism is completely suppressed. Both quantum-critical and renormalized-classical behavior can occur in certain parameter ranges. It is shown that in the 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 upper critical dimension for this phenomenon is three. The analogous phenomenon of pairing pseudogap can occur in the attractive model in two dimensions when the pairing fluctuations become critical. Simple analytical expressions for the self-energy are derived in both the magnetic and pairing pseudogap regimes. Other approaches, such as paramagnon, self-consistent fluctuation exchange approximation (FLEX), and pseudo-potential parquet approaches are critically compared. In particular, it is argued that the failure of the FLEX approximation to reproduce the psuedogap and the precursors AFM bands in the weak coupling regime and the Hubbard bands in the strong coupling regime is due to inconsistent treatment of vertex corrections in the expression for the self-energy. Treating the spin fluctuations as if there was a Migdal's theorem can lead not only to quantitatively wrong results but also to qualitatively wrong predictions, in particular with regard to the single-particle pseudogap.