The theory of composites Some equations of interest and numerical approaches to solving them Duality transformations in two dimensional media Translations and equivalent media Some microstructure independent exact relations Exact relations for coupled equations Assemblages of spheres ellipsoids and other neutral inclusions Tricks for generating other exactly solvable microgeometries Laminate materials Approximations and asymptotic formulas Wave propagation in the quasistatic limit Reformulating the problem of finding effective tensors Variational principles and inequalities Series expansions for the fields and effective tensors Correlation functions and how they enter series expansions Other perturbation solutions The general theory of exact relations and links between effective tensors Analytic properties Y tensors Y tensors and effective tensors in electrical circuits Bounds on the properties of composites Classical variational principle bounds Bounds from the Hashin Shtrikman variational inequalities Bounds using the compensated compactness or translation method Choosing the translations and finding microgeometries that attain the bounds Bounds incorporating three point correlation functions Bounds using the analytic method Fractional linear transformations as a tool for generating bounds The field equation recursion method Properties of the G closure and extremal families of composites The bounding of effective moduli as a quasiconvexification problem Although extensively studied for more than a hundred years an explosion of ideas related to composite materials in the last four decades has dramatically increased our understanding of the relationship between the properties of the constituent materials the underlying microstructure of a composite and the overall effective (electrical thermal elastic) moduli which govern the macroscopic behavior. This book surveys these exciting developments at the frontier of mathematics and presents many new results