Black holes in the Universe do not exist in isolation but, rather, they are surrounded by matter. It is therefore important to study the stability properties of black holes under matter field perturbations. In this talk we will discuss the stability properties under classical field perturbations of several rotating (Kerr) black hole spacetimes. In particular, we will present recent results on some of the mode stability properties of the following: (i) a maximally-rotating (extremal) Kerr black hole, whose event horizon suffers from the so-called Aretakis instability; (ii) Kerr-de Sitter spacetime, representing a Kerr black hole in a Universe with accelerated expansion, and so possessing a cosmological horizon as well as an event horizon; and (iii) the inner (Cauchy) horizon of Kerr-Newman-de Sitter spacetime, representing a charged Kerr-de Sitter black hole, where we find evidence for violation of Penrose’s strong Cosmic Censorship conjecture.