  In Einstein-Maxwell theory, according to classic uniqueness theorems, the
most general stationary black-hole solution is the axisymmetric Kerr-Newman
metric, which is defined by three parameters: mass, spin and electric charge.
The radial and angular dependence of gravitational and electromagnetic
perturbations in the Kerr-Newman geometry do not seem to be separable. In this
paper we circumvent this problem by studying scalar, electromagnetic and
gravitational perturbations of Kerr-Newman black holes in the slow-rotation
limit. We extend (and provide details of) the analysis presented in a recent
Letter [arXiv:1304.1160]. Working at linear order in the spin, we present the
first detailed derivation of the axial and polar perturbation equations in the
gravito-electromagnetic case, and we compute the corresponding quasinormal
modes for any value of the electric charge. Our study is the first
self-consistent stability analysis of the Kerr-Newman metric, and in principle
it can be extended to any order in the small rotation parameter. We find
numerical evidence that the axial and polar sectors are isospectral at first
order in the spin, and speculate on the possible implications of this result.
