  This paper is concerned with a family of second-order elliptic systems in
divergence form with rapidly oscillating periodic coefficients. We initiate the
study of homogenization and boundary layers for Neumann problems with
first-order oscillating boundary data. We identify the homogenized system and
establish the sharp rate of convergence in $L^2$ in dimension three or higher.
Sharp regularity estimates are also obtained for the homogenized boundary data
in both Dirichlet and Neumann problems. The results are used to obtain a
higher-order convergence rate for Neumann problems with non-oscillating data.
