This talk, based on a joint work with V. S. Shulman and L. Turowska, will be centred around the notion of a multiplicity set in locally compact groups and in products of measure spaces. After introducing the appropriate definitions and giving examples, I will present the relation between the Operator Algebraic setting and the Harmonic-Analytic one. I will discuss preservation properties of multiplicity sets and give applications to closable multipliers acting on group C*-algebras and on group von-Neumann algebras.