In this monograph we shall be concerned with a family of second-order linear elliptic operators in divergence form with rapidly oscillating periodic coefficients, Lε = -div A(x/ε)∇, ε>0, (2.0.1) in ℝd. The coefficient matrix (tensor) A in (2.0.1) is given by A(y) = aαβ ij(y), with 1 ≤ i, j ≤ d and 1 ≤ α, β ≤ m. Thus, if u = (u1, u2, …, um), [Formula Presented.] (the repeated indices are summed). We will always assume that A is real, bounded measurable, and satisfies a certain ellipticity condition, to be specified later. We also assume that A is 1-periodic; i.e., for each z ∈ ℤd, A(y + z) = A(y) for a.e. y ∈ ℝd. (2.0.2) Observe that by a linear transformation one may replace ℤd in (2.0.2) by any lattice in ℝd.