  Using pure spinors, the superstring is covariantly quantized. For the first
time, massless vertex operators are constructed and scattering amplitudes are
computed in a manifestly ten-dimensional super-Poincar\'e covariant manner.
Quantizable non-linear sigma model actions are constructed for the superstring
in curved backgrounds, including the $AdS_5\times S^5$ background with
Ramond-Ramond flux.
