A thermodynamic perturbation theory for fluid mixtures composed of rigid, nonspherical molecules is presented. In this theory the important isotropic and anisotropic interactions arising from two-body electrostatic, induction, dispersion and repulsion forces and from three-body induction and dispersion forces are treated as perturbations of the reference system represented by the Lennard-Jones pair interactions. This form of the perturbation theory differs from its previous versions in that the properties of the reference Lennard-Jones system are computed using an accurate form of the perturbation theory. Numerous comparison of theoretical results with experimental data for vapor-liquid phase equilibria and thermodynamic properties of several pure fluids composed of spherical (argon, krypton, xenon), octopolar (methane), quadrupolar (ethylene, ethane, carbon dioxide) and dipolar (hydrogen bromide) molecules show the very good performance of the perturbation theory. Part II of this work reports results for vapor-liquid phase equilibria and thermodynamic excess properties of binary fluid mixtures.