The effects of magnetic dopants on the superconducting state of the host material has been an important topic in the early days of research on superconductivity. The field has experienced a resurgence in late 1990s with the developments in building and probing nano-scale systems, such as magnetic adsorbate structures on surfaces of superconductors, or quantum dots in hybrid semiconductor-superconductor devices. More recently, the physics of sub-gap states received a further impetus due to its relevance for topological bound states such as Majorana zero modes.

I will discuss the paradigmatic case of a quantum dot (described as a single orbital with on-site electron repulsion term) coupled to a superconducting bath (described using the BCS mean-field Hamiltonian), the applications of the numerical renormalization group method which provides reliable and accurate results in the full parameter space of this model, as well as some recent efforts at systematically confronting the theory with the experiments by mapping out the phase diagram of the quantum phase transition between the 0 and pi states which are characterized by different phase-dependencies of the Josephson current. I will briefly mention some extensions to more complex impurity models (high-spin, magnetic anisotropy, multiple channels).