What are the advantages and the troubles of performing an Abelian and a non-Abelian T-duality over a type IIB/IIA solution? I have seen that Maldacena and Alday found some correspondence between the 4-point gluon scattering and the Wilson loop after T-dualised the geometry, for instance. However, I would like to understand better the idea and motivation behind the (Abelian/non-Abelian)T-duality and also the problems or the unclear things. One possible problem is the fact that T-duality is a symmetry at the order of perturbation string theory, why?
I will answer from my experience of Abelian and non-Abelian T-duality (NATD) as a generating technique in Supergravity and its application to AdS/CFT. Certainly it is not the full story, as I have not much knowledge of the string sigma-model side of it. A general introduction to NATD in the Neveu-Schwartz sector can be found in 9410237. The original paper that extended it to the Ramond-Ramond sector is 1012.1320.
I have used T duality as a generating technique. When you apply the T-duality transformation (both Abelian and non-Abelian) rules to a IIA/IIB SUGRA background with Abelian or non-Abelian isometry, you obtain another solution in IIB/IIA respectively. That was the motivation in my studies, but certainly it is not the only. The main interest of dualities in general is the ability to relate string Physics in two different backgrounds, being the canonical instance the relation between two cylinders of radii $R$ and $1/R$ for Abelian duality.
Non-Abelian duality is not known for sure to be a perturbative symmetry of the string partition function, as it was proven for abelian duality in https://arxiv.org/abs/hep-th/9110053. The dual background can be extremely different from the original one, and typically nearly impossible to find in any other manner. Indeed, it is frequent that they fall outside of known classification of BPS solutions (see for a particular instance https://arxiv.org/abs/1507.02659).
Another limitation of non-Abelian T-duality is that the range of the dual coordinates is not prescribed by the technique. Instead, for the Abelian duality to be a symmetry,a particular periodicity of dual coordinate is necessary (again 9110053).
Another important important point is the AdS/CFT effect, because typically the dual field theory is changed, as you can infer from the brane content, and some holographic observables. A nice example of this can be found in 1301.6755 and its many citations. Hope that helps.