  We discuss the band-gap structure and the integrated density of states for
periodic elliptic operators in the Hilbert space $L_2(\R^m)$, for $m \ge 2$. We
specifically consider situations where high contrast in the coefficients leads
to weak coupling between the period cells. Weak coupling of periodic systems
frequently produces spectral gaps or spectral concentration.
  Our examples include Schr\"odinger operators, elliptic operators in
divergence form, Laplace-Beltrami-operators, Schr\"odinger and Pauli operators
with periodic magnetic fields. There are corresponding applications in heat and
wave propagation, quantum mechanics, and photonic crystals.
