This article is divided into two chapters. The
classical problem of homogenization of elliptic operators with
periodically oscillating coefficients is revisited in the
first chapter. Following a Fourier approach, we discuss some
of the basic issues of the subject: main convergence theorem,
Bloch approximation, estimates on second order derivatives,
correctors for the medium, and so on. The second chapter is
devoted to the discussion of some non-classical behaviour of
vibration problems of periodic structures.