In this thesis we prove an integrated local energy decay estimate for axisymmetric solutions to the Teukolsky equation on slowly rotating Kerr exterior backgrounds. We employ physical space transformation theory based on fixed-frequency theory of Chandrasekhar (as was introduced in [12]), resulting in a generalised Regge-Wheeler
type equation. This allows us, under the assumption of axisymmetry, to use entirely physical space methods. This is convenient for potentially proving a non-linear stability result (restricted to axisymmetry) for slowly rotating Kerr spacetimes. We also prove an integrated local energy decay estimate for solutions to the homogeneous generalised Regge-Wheeler equation on Kerr exterior backgrounds for the full subextremal range of Kerr parameters.