  We study the magnetic anisotropy effects on the localized sub-gap excitations
induced by quantum impurities coupled to a superconducting host. We establish
the ground-state phase diagrams for single-channel and two-channel high-spin
Kondo impurities; they unveil surprising complexity that results from the
(multi-stage) Kondo screening in competition with the superconducting
correlations and the magnetic anisotropy splitting of the spin multiplets. We
discuss the possibility of detecting the Zeeman splitting of the sub-gap
states, which would provide an interesting spectroscopic tool for studying the
magnetism on the single-atom level. We also study the problem of two impurities
coupled by the Heisenberg exchange interaction, and we follow the evolution of
the sub-gap states for both antiferromagnetic and ferromagnetic coupling. For
sufficiently strong antiferromagnetic coupling, the impurities bind into a
singlet state that is non-magnetic, thus the sub-gap states move to the edge of
the gap and can no longer be discerned. For ferromagnetic coupling, some
excited states remain present inside the gap.
