  In this paper, we first prove global well-posedness for the defocusing cubic
nonlinear Schr\"odinger equation (NLS) on 4-dimensional tori - either rational
or irrational - and with initial data in $H^1$. Furthermore, we prove that if a
maximal-lifespan solution of the focusing cubic NLS $u: I\times\mathbb{T}^4\to
\mathbb{C}$ satisfies $\sup_{t\in
I}\|u(t)\|_{\dot{H}^1(\mathbb{T}^4)}<\|W\|_{\dot{H}^1(\mathbb{R}^4)}$, then it
is a global solution. $W$ denotes the ground state on Euclidean space, which is
a stationary solution of the corresponding focusing equation in $\mathbb{R}^4$.
