  Recent progress in nanoscale manufacturing allowed to experimentally
investigate quantum dots coupled to two superconducting leads in controlled and
tunable setups. The equilibrium Josephson current was measured in on-chip SQUID
devices and subgap states were investigated using weakly coupled metallic leads
for spectroscopy. This put back two "classic" problems also on the agenda of
theoretical condensed matter physics: the Josephson effect and quantum spins in
superconductors. We discuss the status of the theoretical understanding of the
Anderson-Josephson quantum dot in equilibrium mainly focusing on the Josephson
current. We introduce a minimal model consisting of a dot which can only host
one spin-up and one spin-down electron repelling each other by a local Coulomb
interaction. The dot is tunnel-coupled to two superconducting leads described
by the BCS Hamiltonian. This model was investigated using a variety of methods,
some capturing aspects of Kondo physics others failing in this respect. We
briefly review this. The model shows a first order level-crossing quantum phase
transition when varying any parameter provided the others are within
appropriate ranges. At vanishing temperature it leads to a jump of the
Josephson current. We show that a quantitative agreement between accurate
results obtained for the simple model and measurements of the current can be
reached. This confirms that the experiments reveal the finite temperature
signatures of the zero temperature transition. In addition, we consider two
examples of more complex dot geometries which might be experimentally realized
in the near future. The first is characterized by the interplay of the above
level-crossing physics and the Fano effect, the second by the interplay of
superconductivity and almost degenerate singlet and triplet two-body states.
