Abstract
We present a state-of-the-art review of multi-scale finite element methods : the problems these methods tackle, the development of different approaches, advances in their performance and in their theoretical understanding. We will also highlight the difficulties they may encounter, the mathematical issues open to their analysis, and the practical aspects that may slow down their wider use. The discussion will be illustrated by a sample of work carried out in collaboration with Frédéric Legoll, Alexei Lozinski, Ulrich Hetmaniuk and, in particular, François Madiot and Rutger Biezemans.
