Quasinormal modes of black holes are supposed to describe oscillations and decay of gravitational waves produced by the black hole interacting with another object. They have been extensively studied by physicists, most recently in the context of string theory. We provide a rigorous definition of quasinormal modes of slowly rotating Kerr-de Sitter black holes, resonance expansions of linear waves in terms of these modes (and in particular exponential decay), and a semiclassical description of quasinormal modes, which matches numerical results already at low energies.