In General Relativity (GR) the notion of a Killing symmetry plays a fundamental role, with applications to the construction and characterisation of exact solutions, questions of integrability of the geodesic equations and the construction of Morawetz estimates, to name but a few. During my stay at the ESI, I aim to study two problems which arise naturally in GR. Firstly: is there a sense in which we can say that a spacetime is close to possessing a Killing symmetry? Secondly: what are the necessary and sufficient conditions for the existence of an exact Killing symmetry? In the first project, I hope to extend the notion of approximate KID sets, and their resulting Dain invariants, to spacetimes possessing black hole horizons. In the second project, I will study the tractor calculus of the spatial Killing spinor equation in an attempt to identify necessary conditions for the existence of a Killing spinor candidate.