We derive new, exact series expansions for the effective elastic tensor of anisotropic, d-dimensional, two-phase disordered composites whose nth-order tensor coefficients are integrals involving n-point correlation functions that characterize the structure. These series expansions, valid for any structure, perturb about certain optimal dispersions. Third-order truncation of the expansions results in formulas for the elastic moduli of isotropic dispersions that are in very good agreement with benchmark data, always lie within rigorous bounds, and are superior to popular self-consistent approximations