  We focus on the study of ground-states for the system of $M$ coupled
semilinear Schr\"odinger equations with power-type nonlinearities and
couplings. General results regarding existence and characterization are derived
using a variational approach. We show the usefulness of such a characterization
in several particular cases, including those for which uniqueness of
ground-states is already known. Finally, we apply the results to find the
optimal constant for the vector-valued Gagliardo-Nirenberg inequality and we
study global existence, $L^2$-concentration phenomena and blowup profile for
the evolution system in the $L^2$-critical power case.
