  This chapter presents an overview of the properties of a Bose-Einstein
condensate (BEC) trapped in a periodic potential. This system has attracted a
wide interest in the last years, and a few excellent reviews of the field have
already appeared in the literature (see, for instance, [1-3] and references
therein). For this reason, and because of the huge amount of published results,
we do not pretend here to be comprehensive, but we will be content to provide a
flavor of the richness of this subject, together with some useful references.
On the other hand, there are good reasons for our effort. Probably, the most
significant is that BEC in periodic potentials is a truly interdisciplinary
problem, with obvious connections with electrons in crystal lattices, polarons
and photons in optical fibers. Moreover, the BEC experimentalists have reached
such a high level of accuracy to create in the lab, so to speak, paradigmatic
Hamiltonians, which were first introduced as idealized theoretical models to
study, among others, dynamical instabilities or quantum phase transitions.
