The antiferromagnetic spin correlation function SQ, the staggered spin susceptibility χQ and the energy scale ωF L = SQ/χQ are studied numerically within the t-J model and the Hubbard model, as relevant to cuprates. It is shown that ωF L, related to the onset of the non-Fermi-liquid spin response at T > ωF L, is very low in the regime below the 'optimum' hole doping c h < c * h ∼ 0.16, while it shows a steep increase in the overdoped regime. A quantitative analysis of NMR spin-spin relaxation-rate 1/T2G for various cuprates reveals a similar behavior, indicating on a sharp, but continuous, crossover between a Fermi-liquid and a non-Fermiliquid behavior as a function of doping. PACS numbers: 71.27.+a, The understanding of the phase diagram of cuprates continues to exemplify one of the major theoretical and experimental challenges [1]. Besides superconductivity (SC) and antiferromagnetic (AFM) ordering, several regimes with distinct electronic properties have been identified within the normal metallic phase. The behavior of spin degrees of freedom, which are the subject of this paper, has been intensively studied using the inelastic neutron scattering (INS) [2,3] and NMR relaxation experiments [4]. They clearly reveal that in underdoped cuprates magnetic properties are not following the usual Fermi-liquid (FL) scenario within the metallic state above the SC transition T > T c .Within a normal FL one expects a dynamical spin susceptibility χ ′′ q (ω) to be T -independent at low T, ω. On the contrary, INS results show that q-integrated spin susceptibility exhibits in a broad range of ω and T an anomalous, but universal behavior χ ′′ L (ω) ∝ f (ω/T ), first established in La 2−x Sr x CuO 4 (LSCO) at low doping [3,5]. This behavior can be even followed to lowest T in YBaCu 3 O 6+x (YBCO), where T c has been suppressed by Zn doping [6]. At the same time, low-energy INS reveals at low T the saturation of the inverse AFM correlation length κ = 1/ξ, at least in YBCO [2] and in LSCO at low doping [3,5]. Anomalous T -dependence of 63 Cu NMR spin-lattice relaxation rate 1/T 1 and of the spin-spin relaxation rate 1/T 2G in underdoped cuprates is in general compatible with INS [4], in particularOn the other hand, cuprates at optimum doping and, moreover, in the overdoped regime show a strong reduction of the spin response at low energies ω. This is evident from the loss of INS intensity in the normal state (as well as in a weak resonant peak for T < T c ) and low NMR relaxation rates 1/T 1 , 1/T 2G . At the same time, NMR confirms the approach to the normal FL behavior, 1/(T 1 T ) ∼ const. and 1/T 2G ∼ const. [4]. There are other indications that the normal FL behavior is approached in the overdoped regime. Recently, the angle-resolved photoemission spectroscopy (ARPES) on BiSrCaCuO (BSCCO) system gave evidence for the existence of coherent electronic excitations for T < T X at higher doping [7], i.e., the FL-like phase is found in the normal state only in the overdoped regime where T X shows a steep increase with hole doping...