Electrostatic potentials near the surface of DNA were calculated using the nonlinear Poisson-Boltzmann (NLPB) equation. The results are found to be in good agreement with the potentials measured using ELDOR spectroscopy. In contrast, the linearized PB equation is found to significantly underestimate the screening effects of salt and thus produces potentials that are too negative. Several physical models including a cylinder of radius 10 Angstrom and a detailed atomic structure built from crystallographic coordinates were used to examine the effect of local DNA structure on the calculated potentials. Two dielectric models for the solvent were used; one in which the solvent has a uniform dielectric of 80, and one in which a 5.6-Angstrom layer of solvent at the DNA surface was assigned a dielectric of 10 to simulate dielectric saturation. We find no evidence for dielectric saturation; indeed, including such effects reduces the agreement between theory and experiment. Finally, a simple model in which the charges on the phosphates have been reduced to a value of 0.24 and the potentials are given by Debye-Huckel theory is shown to give a realistic representation of the electrostatic potentals near the surface of DNA.