Quantization of the superstring in Ramond-Ramond backgrounds

Sigma model actions are constructed for the type II superstring compactifie d to four-and six-dimensional curved backgrounds which can contain non-vani shing Ramond-Ramond fields. These actions are N = 2 worldsheet superconform ally invariant and can be covariantly quantized preserving manifest spaceti me supersymmetry. They are constructed using a hybrid Version of superstrin g variables which combines features of the Ramond-Neveu-Schwarz and Green-S chwarz formalisms. For the AdS(2) x S-2 and AdS(3) x S-3 backgrounds, these actions differ from the classical Green-Schwarz actions by a crucial kinet ic term for the fermions. Parts of this work have been done in collaboratio ns with M Bershadsky, T Hauer, W Siegel, C Vafa, E Witten, S Zhukov and B Z wiebach.