Integrable deformations of AdS superstrings and Weyl invariance

I will discuss recent developments relating the Weyl invariance (or lack thereof) to the algebraic structure underlying certain integrable deformations of AdS superstrings. The talk will be focused on two classes of models, the homogeneous and inhomogeneous Yang-Baxter deformations. In the first part I will outline how homogeneous Yang-Baxter deformations are related to non-abelian duality and discuss the consequences for the Weyl invariance of the deformed models. In the second part I will comment on the extent to which these results can be extended to the inhomogeneous Yang-Baxter deformation, a.k.a. the eta or q deformation.