  This is a doctoral thesis on the application of techniques originally
developed in the programme of characterisation of supersymmetric solutions to
Supergravity theories, to finding alternative backgrounds. We start by
discussing the concept of a Killing spinor, and how these are paramount to the
process of classifying the aforementioned supersymmetric solutions. Moreover,
these geometric objects also have applications when considered in different
scenarios (the 'beyond' in the title). In particular, techniques based on a
different parallelising rule for a spinorial field can be used for obtaining
solutions to Einstein-Maxwell-De Sitter theories, as well as a (partial)
classification of lorentzian Einstein-Weyl manifolds, a problem of geometrical
interest. The annexe contain an introduction and summary in Spanish language.
The appendices discuss the tensorial and spinorial conventions employed, some
relevant geometrical information on the scalar manifolds for the matter
contents of interest, as well as for the null case solutions, including Kundt
waves. Also, a short introduction to Weyl geometry and the Similitude group, as
well as the Lorentz and Spin groups, is given.
