  In this paper, we study cubic and quintic nonlinear Schr\"odinger systems on
3-dimensional tori, with initial data in an adapted Hilbert space
$H^s_{\underline{\lambda}},$ and all of our results hold on rational and
irrational rectangular, flat tori. In the cubic and quintic case, we prove
local well-posedness for both focusing and defocusing systems. We show that
local solutions of the defocusing cubic system with initial data in
$H^1_{\underline{\lambda}}$ can be extended for all time. Additionally, we
prove that global well-posedness holds in the quintic system, focusing or
defocusing, for initial data with sufficiently small
$H^1_{\underline{\lambda}}$ norm. Finally, we use the energy-Casimir method to
prove the existence and uniqueness, and nonlinear stability of a class of
stationary states of the defocusing cubic and quintic nonlinear Schr\"odinger
systems.
