Abstract
We review the developments of a recently proposed approach to study
integrable theories in any dimension. The basic idea consists in generalizing
the zero curvature representation for two-dimensional integrable models to
space-times of dimension $d+1$ by the introduction of a $d$-form connection.
The method has been used to study several theories of physical interest, like
self-dual Yang-Mills theories, Bogomolny equations, non-linear sigma models and
Skyrme-type models. The local version of the generalized zero curvature
involves a Lie algebra and a representation of it, leading to a number of
conservation laws equal to the dimension of that representation. We discuss the
conditions a given theory has to satisfy in order for its associated zero
curvature to admit an infinite dimensional (reducible) representation. We also
present the theory in the more abstract setting of the space of loops, which
gives a deeper understanding and a more simple formulation of integrability in
any dimension.