  The most general stationary black-hole solution of Einstein-Maxwell theory in
vacuum is the Kerr-Newman metric, specified by three parameters: mass M, spin J
and charge Q. Within classical general relativity, the most important and
challenging open problem in black-hole perturbation theory is the study of
gravitational and electromagnetic fields in the Kerr-Newman geometry, because
of the indissoluble coupling of the perturbation functions. Here we circumvent
this long-standing problem by working in the slow-rotation limit. We compute
the quasinormal modes up to linear order in J for any value of Q and provide
the first, fully-consistent stability analysis of the Kerr-Newman metric. For
scalar perturbations the quasinormal modes can be computed exactly, and we
demonstrate that the method is accurate within 3% for spins J/Jmax<~0.5, where
Jmax is the maximum allowed spin for any value of Q. Quite remarkably, we find
numerical evidence that the axial and polar sectors of the
gravito-electromagnetic perturbations are isospectral to linear order in the
spin. The extension of our results to nonasymptotically flat space-times could
be useful in the context of gauge/gravity dualities and string theory.
