The spin dynamics in single crystals of YBa 2Cu 3O 6 + x has been successfully investigated, by inelastic neutron scattering (INS) experiments, as a function of temperature in the metallic state over the whole doping range from the weakly doped ( x = 0.45, 0.51) to the heavily doped ( x = 0.69, 0.83, 0.92) and the overdoped ( x = 0.97, 1) regimes. Dynamical AF-correlations persist in all the metallic state. Actually, the imaginary part of the magnetic susceptibility χ″ consists of two contributions which have different doping and temperature dependences. At low temperature, χ″ exhibits an energy gap in any superconducting samples which becomes much weaker close to the insulating-metallic transition. To emphasize the characteristic features of the spin dynamics in YBCO, INS results obtained by other groups are compared with our experiments. Several theoretical approaches, which intend to describe the energy line shape of the dynamical magnetic susceptibility, are also discussed.