  In this Thesis we study first- and second-quantized approaches describing
$D=11$ supergravity using pure spinor variables. We introduce the so-called
$D=11$ pure spinor superparticle through BRST cohomology arguments starting
from the semi-light-cone gauge $D=11$ Brink-Schwarz-like superparticle. After
performing a light-cone gauge analysis of the pure spinor BRST cohomology at
ghost number three, we find the linearized equations of motion of $D=11$
supergravity in $D=9$ superspace. In addition, we construct a BRST-closed,
ghost number one vertex operator made out of worldline fields and $D=11$
supergravity superfields, and we run into an inconsistency when constructing a
ghost number zero vertex operator satisfying a standard descent equation. We
then introduce the non-minimal version of the $D=11$ pure spinor superparticle,
in which a composite $b$-ghost can be constructed satisfying $\{Q,b\} = P^{2}$.
However, its complicated expression makes it difficult to check its nilpotency.
We show that introducing an $SO(1,10)$ fermionic vector $\bar{\Sigma}^{a}$
simplifies the form of the $b$-ghost considerably, which allows us to verify
that $\{Q,b\} = P^{2}$ and $\{b,b\}=$ BRST-exact. Using this $b$-ghost we
propose an alternative ghost number zero vertex operator satisfying a standard
descent equation. However, its expression will depend on non-minimal pure
spinor variables in a very complicated fashion. After discussing this
first-quantized approach for $D=11$ supergravity, we move on to discussing the
pure spinor master actions introduced by Cederwall for studying maximally
supersymmetric gauge theories. We show that these actions indeed describe
$D=10$ super-Yang-Mills, $D=10$ super-Born-Infeld and $D=11$ supergravity by
extracting the equations of motion in ordinary superspace for each one of these
theories.
