It is shown that for many composites an accurate method to predict elastic moduli is to average the upper and lower (third-order) bounds. Good agreement with experiment is obtained for the metal-matrix composite Al/SiC. The elastic properties of random fiber-aligned composites have been calculated using a digital-image-based simulation with application to the glass fiber/epoxy system and to rigid inclusions. Comparison to Hashin-Shtrikman and third-order bounds has been made. In three dimensions, the finite-element method has been used to study ordered composites. It has been found that for rigid inclusions (voids), the lower (upper) bound on the bulk modulus provides a good estimate except near the critical composition, φc, where the moduli for ordered rigid particles diverge as (φ-φc)-1/2 in 2D and as -ln(φ-φc) in 3D.