  In this chapter we describe a selection of mathematical techniques and
results that suggest interesting links between the theory of gratings and the
theory of homogenization, including a brief introduction to the latter. By no
means do we purport to imply that homogenization theory is an exclusive method
for studying gratings, neither do we hope to be exhaustive in our choice of
topics within the subject of homogenization. Our preferences here are motivated
most of all by our own latest research, and by our outlook to the future
interactions between these two subjects. We have also attempted, in what
follows, to contrast the "classical" homogenization (Section 11.1.2), which is
well suited for the description of composites as we have known them since their
advent until about a decade ago, and the "non-standard" approaches,
high-frequency homogenization (Section 11.2) and high-contrast homogenization
(Section 11.3), which have been developing in close relation to the study of
photonic crystals and metamaterials, which exhibit properties unseen in
conventional composite media, such as negative refraction allowing for
super-lensing through a flat heterogeneous lens, and cloaking, which
considerably reduces the scattering by finite size objects (invisibility) in
certain frequency range. These novel electromagnetic paradigms have renewed the
interest of physicists and applied mathematicians alike in the theory of
gratings.
