"The proposal aims at investigating several aspects of the recently proposed ""pure spinor"" formalism for the quantization of the superstring in a way that keeps all the symmetries manifest. The problem we intend to study has become more and more pressing in recent years, because of the importance and relevance of string theory backgrounds containing fluxes for Ramond-Ramond fields. They appear in the context of gauge/gravity dualities, in the string description of black holes and in cosmological applications, namely some of the most promising directions in which string theory is progressing. A detailed quantitative understanding of the properties of interest in all these cases crucially depends on the knowledge of the interactions with the RR fields, and the pure spinor formalism is the only one capable of addressing the question. On the other hand, the construction of the formalism is still to some extent work-in-progress and has several open issues. Our proposal is divided in two main aspects: a more formal one, oriented to studying the issues involved with a proper definition of the model; this requires the study of a more general class of models (beta-gamma systems, and sigma models on supergroups) that are also of independent interest in the broader context of conformal field theory; and a more concrete and computational aspect, oriented to the calculation of the corrections to the effective string dynamics that depend on the presence of the RR fields."