  In this thesis are presented two aplications of the sigma model for the
superstring in the pure spinor formulation. The first aplication concerns the
computation of the one-loop conformal invariance for the type II superstring,
resulting in equations of motion written in superspace for the background
fields coupled to the superstring. The second application is related to the
BRST invariance of the heterotic superstring at the quantum level, which allows
to find stringy corrections to the ten-dimensional super
Yang-Mills/supergravity constraints.
