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1. Introduction 2. Equations of interest and numerical approaches 3. Duality transformations 4. Translations and equivalent media 5. Microstructure independent exact relations 6. Exact relations for coupled equations 7. Assemblages of inclusions 8. Tricks for exactly solvable microgeometries 9. Laminate materials 10. Approximations and asymptotic formulae 11. Wave propagation in the quasistatic limit 12. Reformulating the problem 13. Variational principles and inequalities 14. Series expansions 15. Correlation functions and series expansions 16. Other perturbation solutions 17. The general theory of exact relations 18. Analytic properties 19. Y-tensors 20. Y-tensors and effective tensors in circuits 21. Bounds on the properties of composites 22. Classical variational principle bounds 23. Hashin-Shtrikman bounds 24. Translation method bounds 25. Choosing translations and finding geometries 26. Bounds incorporating three-point statistics 27. Bounds using the analytic method 28. Fractional linear transformations for bounds 29. The field equation recursion method 30. G-closure properties and extremal composites 31. Bounding and quasiconvexification.