  We construct electrically charged Kerr black holes (BHs) with scalar hair.
Firstly, we take an uncharged scalar field, interacting with the
electromagnetic field only indirectly, via the background metric. The
corresponding family of solutions, dubbed Kerr-Newman BHs with ungauged scalar
hair, reduces to (a sub-family of) Kerr-Newman BHs in the limit of vanishing
scalar hair and to uncharged rotating boson stars in the limit of vanishing
horizon. It adds one extra parameter to the uncharged solutions: the total
electric charge. This leading electromagnetic multipole moment is unaffected by
the scalar hair and can be computed by using Gauss's law on any closed
2-surface surrounding (a spatial section of) the event horizon. By contrast,
the first sub-leading electromagnetic multipole -- the magnetic dipole moment
--, gets suppressed by the scalar hair, such that the gyromagnetic ratio is
always smaller than the Kerr-Newman value ($g=2$). Secondly, we consider a
gauged scalar field and obtain a family of Kerr-Newman BHs with gauged scalar
hair. The electrically charged scalar field now stores a part of the total
electric charge, which can only be computed by applying Gauss' law at spatial
infinity and introduces a new solitonic limit -- electrically charged rotating
boson stars. In both cases, we analyse some physical properties of the
solutions.
