Periodic Homogenization of Elliptic Systems This monograph surveys the theory of quantitative homogenization for second order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients in a bounded domain. It begins with a review of the classical qualitative homogenization theory and addresses the problem of convergence rates of solutions. The main body of the monograph investigates various interior and boundary regularity estimates that are uniform in the small parameter e>0. Additional topics include convergence rates for Dirichlet eigenvalues and asymptotic expansions of fundamental solutions Green functions and Neumann functions. The monograph is intended for advanced graduate students and researchers in the general areas of analysis and partial differential equations. It provides the reader with a clear and concise exposition of an important and currently active area of quantitative homogenization Elliptic Systems of Second Order with Periodic Coeffcients Convergence Rates Part I Interior Estimates Regularity for Dirichlet Problem Regularity for Neumann Problem Convergence Rates Part II L2 Estimates in Lipschitz Domains