We have performed a comprehensive investigation on the spin susceptibility of the organic quantum-spin systems (TMTTF)(2)X (X = SbF6, AsF6, BF4, ReO4, and SCN) in the temperature range from T = 1.8 up to 380 K. At elevated temperatures, the spin susceptibility at constant volume (chi(s)) V can be described by a spin-1/2 antiferromagnetic Heisenberg chain with exchange constants J between 400 and 460 K. Below T approximate to 100 K, slight deviations from the model occur due to interchain interaction. At low temperatures, the compounds undergo transitions to various ordered states of structural or magnetic origin that are discussed in detail. The spin-dimerized ground states of (TMTTF)(2)AsF6 and (TMTTF)(2)BF4 can be described by an alternating spin chain with a singlet-triplet energy gap Delta(sigma) (0) = 34.8 and 52 K, respectively. In (TMTTF)(2)ReO4, however, the spin susceptibility in the anion-ordered state deviates from this model and obeys an activated lawwith Delta(sigma) = 1100 K. In the antiferromagnetic ground states of (TMTTF)(2)SbF6 and (TMTTF)(2)SCN an increase of the spin susceptibility is observed at the lowest temperatures.