  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.
