  The Petrov classification is an important algebraic classification for the
Weyl tensor valid in 4-dimensional space-times. In this thesis such
classification is generalized to manifolds of arbitrary dimension and
signature. This is accomplished by interpreting the Weyl tensor as a linear
operator on the bundle of p-forms, for any p, and computing the Jordan
canonical form of this operator. Throughout this work the spaces are assumed to
be complexified, so that different signatures correspond to different reality
conditions, providing a unified treatment. A higher-dimensional generalization
of the so-called self-dual manifolds is also investigated.
  The most important result related to the Petrov classification is the
Goldberg-Sachs theorem. Here are presented two partial generalizations of such
theorem valid in even-dimensional manifolds. One of these generalizations
states that certain algebraic constraints on the Weyl operator imply the
existence of an integrable maximally isotropic distribution. The other version
of the generalized Goldberg-Sachs theorem states that these algebraic
constraints imply the existence of a null congruence whose optical scalars obey
special restrictions.
  On the pursuit of these results the spinorial formalism in 6 dimensions was
developed from the very beginning, using group representation theory. Since the
spinors are full of geometric significance and are suitable tools to deal with
isotropic structures, it should not come as a surprise that they provide a
fruitful framework to investigate the issues treated on this thesis. In
particular, the generalizations of the Goldberg-Sachs theorem acquire an
elegant form in terms of the pure spinors.
