  These ICTP Trieste lecture notes review the pure spinor approach to
quantizing the superstring with manifest D=10 super-Poincare invariance. The
first section discusses covariant quantization of the superparticle and gives a
new proof of equivalence with the Brink-Schwarz superparticle. The second
section discusses the superstring in a flat background and shows how to
construct vertex operators and compute tree amplitudes in a manifestly
super-Poincare covariant manner. And the third section discusses quantization
of the superstring in curved backgrounds which can include Ramond-Ramond flux.
