This paper is concerned with two aspects of the theory of the decay of g(r), the radial distribution function of a liquid. For models in which the attractive interatomic potential is short ranged, asymptotic decay falls generically into two classes: (a) monotonic decay for which r(g(r)-1) approximately exp(- alpha ₀r) and (b) damped oscillatory decay for which this function approximately exp(- alpha ₀r)cos( alpha ₁r- theta ). Crossover between the two classes ( alpha ₀= alpha ₀) defines the Fisher-Widom line of the particular model. This line is calculated for a truncated Lennard-Jones fluid using an accurate (HMSA) integral-equation theory. We find that it intersects the liquid branch of the liquid-vapour coexistence curve at T/T_c approximately=0.9 and rho / rho _c approximately=1.9, where T_c and rho _c are the critical temperature and density, respectively. The location of the line relative to coexistence is very similar to that calculated earlier using the random phase approximation (RPA) for a square-well fluid, suggesting that in this region it is not particularly sensitive to choice of potential or of theory. In the second part of the paper we develop a theory for the intermediate-range and asymptotic decay of g(r) for a fluid whose potential includes power-law (dispersion) contributions. Although power-law decay dominates at longest range, we show that intermediate-range oscillatory structure is determined by a single complex pole. Explicit calculations, within the RPA, for a model potential with a 1/r⁶ tail show that at high densities this pole is located close to that of a reference model with a short-ranged truncated potential and the intermediate- and short-range structure of the two models is almost identical. However, since there is no pure imaginary pole for the long-ranged potential, there is no pure exponential decay of correlations and, therefore, no sharply defined Fisher-Widom line. Intermediate-range oscillations in g(r) are eroded at lower densities but the mechanism is different from that in the short-ranged models. In addition, we find that the pole structure of models with large truncation lengths is very different from that of the full potential making asymptotic analysis for such models of little practical use.