Composite materials are treated as microinhomogeneous media. Since their mechanical and ultimate-strength characteristics are determined in large part both by the interaction at the boundaries between the components and by the properties of the intermediate (interphase) layers formed in the manufacture of these materials, prediction of their behavior requires the use of structural micromechanical models and the development of specialized numerical methods. Especially serious problems arise in analysis of the properties of composites that incorporate polymers and exhibit elastoplastic, viscoelastic, or mixed types of behavior, depending on the type of deformation, the temperature, and other factors. Using certain model representations of polymer-composite microstructure and, among others, the hypothesis that interphase layers exist at the matrix-filler boundary, as well as a specialized program package for the finite element method (this journal, Vol. I, No. 3, 1996), we performed computer experiments to model the mechanical behavior and properties of these systems. Calculations of the state of stress and strain and the stress-strain curves of model media based on an epoxy polymer and carbon or glass fibers using a model of elastoplastic flow with a Drucker-Prager strength criterion brought out global differences in the mechanical behavior of these microinhomogeneous polymeric media that depended on the “quality” of the interphase layer. Effective stress-strain characteristics were estimated. It is shown that the macromechanical characteristics of such systems can be predicted with acceptable objectivity only if their microstructural and micromechanical properties are taken into account.