In this talk I will consider linear perturbations of Kerr spacetimes. I will first formulate a linear stability result for these spacetimes and explain the principal ingredients of the proof. I will then show which kind of decay estimates can be obtained for the Teukolsky equation on Kerr spacetimes and how this equation can be quantized. All results are valid in the full subextremal parameter range. The talk is based on different joint works with Peter Hintz, András Vasy (linear stability); Lars Andersson, Bernard Whiting (mode analysis) and Christiane Klein (quantization of Teukolsky) as well as on results of my former student Pascal Millet (decay for Teukolsky).