BPS quivers are graphs encoding the supersymmetric quantum mechanics on the worldlines of 4D N=2 BPS states. These objects have a wide variety of magical properties and interesting applications, relating aspects of 4D N=2 physics with the representation theory of a special class of basic algebras. In this talk, I am going to discuss some applications of the BPS quivers to the problem of classifying 4D N=2 superconformal field theories.