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Using the homogeneous G\"{o}del spacetimes we find some new solutions for thefield equations of bosonic string effective action up to first order in$\alpha'$ including both dilaton and axion fields. We then discuss in detailthe (non-)Abelian T-dualization of G\"{o}del string cosmologies via thePoisson-Lie (PL) T-duality approach. In studying Abelian T-duality of themodels we get seven dual models in such a way that they are constructed byone-, two- and three-dimensional Abelian Lie groups acting freely on the targetspace manifold. The results of our study show that the Abelian T-dual modelsare, under some of the special conditions, self-dual; moreover, by applying theusual rules of Abelian T-duality without further corrections, we are still ableto obtain two-loop solutions. We also study the Abelian T-duality of G\"{o}delstring cosmologies up to $\alpha'$-corrections by using the T-duality rules attwo-loop order derived by Kaloper and Meissner. Afterwards, non-Abelian dualsof the G\"{o}del spacetimes are constructed by two- and three-dimensionalnon-Abelian Lie groups such as $A_2$, $A_2 \oplus A_1$ and $SL(2, \mathbb{R})$.In this way, the PL self-duality of $AdS_3 \times \mathbb{R}$ space isdiscussed.

T-dualization of G\"{o}del string cosmologies via Poisson-Lie T-duality approach