  These notes record three lectures given at the workshop "Higher symmetries in
Physics", held at the Universidad Complutense de Madrid in November 2008. In
them we explain how to construct a Lie (super)algebra associated to a spin
manifold, perhaps with extra geometric data, and a notion of privileged
spinors. The typical examples are supersymmetric supergravity backgrounds;
although there are more classical instances of this construction. We focus on
two results: the geometric constructions of compact real forms of the simple
Lie algebras of type B_4, F_4 and E_8 from S^7, S^8 and S^15, respectively; and
the construction of the Killing superalgebra of eleven-dimensional supergravity
backgrounds. As an application of this latter construction we show that
supersymmetric supergravity backgrounds with enough supersymmetry are
necessarily locally homogeneous.
