I review most of the recent developments in the covariant quantization of superstring theory. First, I discuss some basic elements of the pure spinor formalism for superstrings and for the related theories as superparticles. I will then describe the supergeometry related to the superspace formulation of string theory and I will show that there is an intrinsic definition of “picture” for superforms. Then, I will describe how this can be used to give a prescription for all-order computations in pure spinor string theory. Applications to super Chern-Simons theory viewed as string theory are also discussed.

YITP, Stony Brook and Piem. Orien. U