Integrability of type II superstrings on Ramond-Ramond backgrounds in various dimensions

We consider type II superstrings on AdS backgrounds with Ramond-Ramond flux in various dimensions. We realize the backgrounds as supercosets and analyze explicity two classes of models: non-critical superstrings on AdS(2d) and critical superstrings on AdS(p) x S-p x CY. We work both in the Green-Schwarz and in the pure spinor formalisms. We construct a one-parameter family of flat currents (a Lax connection), leading to an infinite number of conserved non-local charges, which imply the classical integrability of both sigma-models. In the pure spinor formulation, we use the BRST symmetry to prove the quantum integrability of the sigma-model. We discuss how classical kappa-symmetry implies one-loop conformal invariance. We consider the addition of space-filling D-branes to the pure spinor formalism.