  These notes follow from a course delivered at the V Jos\'e Pl\'{\i}nio
Baptista School of Cosmology, held at Guarapari (Esp\'{\i}rito Santo) Brazil,
from 30 September to 5 October 2021. A review of the current status of the
linear stability of black holes and naked singularities is given. The standard
modal approach, that takes advantage of the background symmetries and analyze
separately the harmonic components of linear perturbations, is briefly
introduced and used to prove that the naked singularities in the Kerr--Newman
family, as well as the inner black hole regions beyond Cauchy horizons, are
unstable and therefore unphysical. The proofs require a treatment of the
boundary condition at the timelike boundary, which is given in detail. The
nonmodal linear stability concept is then introduced, and used to prove that
the domain of outer communications of a Schwarzschild black hole with a
non-negative cosmological constant satisfies this stronger stability condition,
which rules out transient growths of perturbations, and also to show that the
perturbed black hole settles into a slowly rotating Kerr black hole. The
encoding of the perturbation fields in gauge invariant curvature scalars and
the effects of the perturbation on the geometry of the spacetime is discussed.
