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We show new global well-posedness results for mass-critical nonlinearSchr\"odinger equations on tori in one and two dimensions. For the quinticnonlinear Schr\"odinger equation on the circle we show global well-posednessfor initial data in $H^s(\mathbb{T})$ for $s>\frac{1}{3}$ and $\| u_0\|_{L^2(\mathbb{T})} \ll 1$. In two dimensions we show global well-posedness onpossibly irrational tori for $s>\frac{3}{5}$. In the focusing case we need asmallness condition for the mass, whereas in the defocusing case large mass iscovered.

Improved global well-posedness for mass-critical nonlinear Schr\"odinger equations on tori