An introduction to the covariant quantization of superstrings

We give an introduction to a new approach to the covariant quantization of superstrings. After a brief review of the classical Green-Schwarz superstring and Berkovits' approach to its quantization based on pure spinors, we discuss our covariant formulation without pure spinor constraints. We discuss the relation between the concept of grading, which we introduced to define vertex operators, and homological perturbation theory, and we compare our work with recent work by others. In the appendices, we include some background material for the Green-Schwarz and Berkovits formulations, in order that this presentation be self-contained.