The recent development of a microscopic theory of the equilibrium properties of polymer solutions, melts and alloys based on off-lattice Polymer Reference Interaction Site Model (PRISM) integral equation methods is reviewed. Analytical and numerical predictions for the intermolecular structure and collective density scattering patterns of both coarse-grained and atomistic models of polymer melts are presented and found to be in good agreement with large scale computer simulations and diffraction measurements. The general issues and difficulties involved in the use of the structural information to compute thermodynamic properties are reviewed. Detailed application of a hybrid PRISM approach to calculate the equation-of-state of hydrocarbon fluids is presented and found to reproduce accurately experimental PVT data on polyethylene. The development of a first principles off-lattice theory of polymer crystallization based on a novel generalization of modern thermodynamic density functional methods is discussed. Numerical calculations for polyethylene and polytetrafluoroethylene are in good agreement with the experimental melting temperatures and liquid freezing densities. Generalization of the PRISM approach to treat phase separating polymer blends is also discussed in depth. The general role of compressibility effects in determining small angle scattering patterns, the effective chi-parameter, and spinodal instability curves are presented. New theoretical concepts and closure approximations have been developed in order to describe correctly long wavelength concentration fluctuations in macromolecular alloys. Detailed numerical and analytical applications of the PRISM theory to model athermal and symmetric blends are presented, and the role of nonmean field fluctuation processes are established. Good agreement between the theory and computer simulations of simple symmetric polymer blends has been demonstrated. Strong, nonadditive compressibility effects are found for structurally and/or interaction asymmetric blends which have significant implications for controlling miscibility in polymer alloys. Recent generalizations of PRISM theory to treat block copolymer melts, and nonideal conformational perturbations, are briefly described. The paper concludes with a brief summary of ongoing work and fertile directions for future research.
