  In recent years considerable advances have been made in quantitative
homogenization of partial differential equations in the periodic and
non-periodic settings. This monograph surveys the theory of quantitative
homogenization for second-order linear elliptic systems in divergence form with
rapidly oscillating periodic coefficients, $$ \mathcal{L}_\e =-\text{\rm div}
\big (A(x/\e)\nabla \big), $$ in a bounded domain $\Omega$ in $\br^d$. 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
(H\"older, Lipschitz, $W^{1, p}$, nontangnetial-maximal-function) 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.
  Part of this monograph is based on lecture notes for courses the author
taught at several summer schools and at the University of Kentucky. Much of
material in Chapters 6 and 7 is taken from his joint papers \cite{KLS-2013,
KLS-2014} with Carlos Kenig and Fang-Hua Lin, and from \cite{KS-2011-L} with
Carlos Kenig.
