The study of wave propagation on black hole spacetimes has been an intense field of research over the last decades. This interest is driven by the stability problem for black holes and by scattering questions. In the case of Maxwell's equations and the equations of linearized gravity on Kerr, it is possible to base the analysis on the study of the Teukolsky equation, which has the advantage of being scalar in nature. I will present a result providing the large time leading-order term for initially localized and regular solutions and valid for the full subextremal range of black hole parameters. I will also present some aspects of the proof which relies on spectral and microlocal methods.