  The pure spinor formalism for the superstring has the advantage over the more
conventional Ramond-Neveu-Schwarz formalism of being manifestly spacetime
supersymmetric, which simplifies the computation of multiparticle and multiloop
amplitudes and allows the description of Ramond-Ramond backgrounds. In addition
to the worldsheet variables of the Green-Schwarz-Siegel action, the pure spinor
formalism includes bosonic ghost variables which are constrained spacetime
spinors and are needed for covariant quantization using a nilpotent BRST
operator.
  In this review, several applications of the formalism are described including
the explicit computation in D=10 superspace of the general disk amplitude with
an arbitrary number of external massless states, genus one amplitudes with up
to seven external states, genus two amplitudes with up to five external states,
and the low-energy limit of the genus three amplitude with up to four external
states. The pure spinor formalism has also been used to covariantly quantize
the superstring in an $AdS_5\times S^5$ background and might be useful for
proving the AdS-CFT correspondence in the limit of small AdS radius.
  This is an overview written for the "Handbook of Quantum Gravity", eds. C.
Bambi, L. Modesto and I. Shapiro.
