The strong salt dependence of the binding of multivalent ligands to DNA indicates the predominantly electrostatic nature of that binding. To a good first approximation, the salt dependence can be expressed as a competition between two point counterion species in screening the highly charged macroion surface. This basic problem is solved in terms of the Poisson-Boltzmann (P-B) equation for a macroion of arbitrary shape and charge, in a solution of arbitrary salt composition and ionic strength. Various binding regimes are defined by the relative magnitudes of a, the radius of curvature of the macroion; rd, the Debye-Hückel screening length, which characterizes exponential decay in the linear screening regime; and λ, the decay length of the counterion distribution close to the macroion surface in the nonlinear screening regime. Only the nonlinear regime (λ ≪ rd) produces relatively high free energy of ligand binding (>kBT per ion). This regime is very similar in planar and cylindrical geometries. For this case we suggest a simple new method of calculating the amount of each species bound, which avoids numerical solution of the P-B equation for each set of species concentrations and further integration of the charge. Instead, we follow changes in the surface concentration and the decay length of each counterion distribution in the course of titration. The product of these two quantities yields a reasonably accurate estimate for the amounts bound. It also yields a closed form of the binding isotherm in both geometries. The apparent ligand (species 2) binding constant obtained in this way has a conventional dependence on the salt (species 1) concentration to the -z2/z1 power. However, we obtain a new expression for the magnitude of the electrostatic binding constant in terms of the macroion surface charge density, Bjerrum length, and ion charges z1 and z2. Binding of counterions to the DNA double helix changes qualitatively from nonlinear cylindrical, through nonlinear planar, to linear planar behavior as the ionic strength of the solution is raised. Our results are comparable in many ways to those of counterion condensation theory and the McGhee-von Hippel site binding model, but show some different behavior and suggest different interpretations of similar behavior.