The inverse-square interaction appears in a wide variety of physical systems  including molecular physics, anyons, graphene with a Coulomb defect, Calogero type integrable models, near-horizon dynamics of non-extremal black holes and certain aspects of holography. The common feature of these systems is an underlying conformal quantum mechanics and the associated symmetry algebra. In this talk we shall discuss certain aspects of the conformal quantum mechanics with particular emphasis on the conformal anomaly arising from the requirement of a unitary time evolution. We shall also discuss the idea of renormalization in these systems and its implication for the famous "fall to the centre problem” originally discussed by Landau.