  We discuss the formation of scalar clouds on charged, spherically symmetric
and static stars and black holes. We first discuss Reissner-Nordstr\"om black
holes with electric and magnetic charge and present new results demonstrating
the existence of a second branch of solutions. Moreover, we find that the
presence of the magnetic charge allows for smaller mass and Noether charge when
the clouds are close to their minimal possible mean radius. Replacing standard
electrodynamics by Born-Infeld (BI) electrodynamics, the background model
possesses globally regular, star-like as well as black hole solutions. We show
that the former can be scalarized as well and that the scalar clouds become
more compact when decreasing the electric charge of the BI star. Finally, we
discuss a magnetic dipole field in the background of the scalarized BI star and
show that the strong gravitational field of this star leads to a significant
change.
