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The double-inclusion model consists of an ellipsoidal inclusion of arbitrary elasticity, containing another ellipsoidal heterogeneity of arbitrary elasticity, size, and orientation, which are embedded in an infinitely extended homogeneous domain of yet another arbitrary elasticity. Average field quantities for the double inclusion are obtained analytically, and used to estimate the overall moduli of two-phase composites. The technique includes the self-consistent and other related methods as special cases. Furthermore, exact bounds for the overall moduli are obtained on the basis of the double-inclusion model. The double-inclusion model has been generalized (Nemat-Nasser and Hori, 1993) to a multi-inclusion model, where, again, all the average field quantities are estimated analytically. The application of the multiinclusion model includes a composite containing inclusions with multi-layer coatings.