  We report on a recent result concerning the effective dynamics for a mixture
of Bose-Einstein condensates, a class of systems much studied in physics and
receiving a large amount of attention in the recent literature in mathematical
physics; for such models, the effective dynamics is described by a coupled
system of non-linear Sch\"odinger equations. After reviewing and commenting our
proof in the mean field regime from a previous paper, we collect the main
details needed to obtain the rigorous derivation of the effective dynamics in
the Gross-Pitaevskii scaling limit.
