Integrable strings for AdS/CFT

In this PhD thesis we review some aspects of integrable models related to string backgrounds or their deformations. In the first part we develop methods to obtain exact results in the AdS3/CFT2 correspondence. We consider the AdS_3 x S^3 x T^4 background with pure Ramond-Ramond flux and we find the all-loop worldsheet S-matrix by exploiting the symmetries of the model in light-cone gauge. As we naturally include the massless modes on the worldsheet, we derive the full set of Bethe-Yang equations. In the massive sector we give also a spin-chain description and we write down solutions compatible with crossing for the scalar factors which are not constrained by simmetries. In the second part of the thesis we consider the so-called "eta-deformation" of the superstring on AdS_5 x S^5. We first discuss the effects of the deformation at the level of the bosonic sigma-model, and we match the tree-level worldsheet scattering processes to the expansion of the q-deformed S-matrix. To identify the missing Ramond-Ramond fields we then compute the action quadratic in fermions, and we provide an alternative derivation by looking at the kappa-symmetry variations. The resulting background fields do not solve the equations of motion of type IIB supergravity and we comment on this