Abstract: "There is a classical mathematical theorem (based on the work by Banach and Tarski) that implies the following shocking statement: “An orange can be divided into finitely many pieces, these pieces can be moved and rearranged in such a way to yield two oranges of the same size as the original one.” In 1929 J. von Neumann recognized that one of the reasons underlying the Banach-Tarski paradox is the fact that on the unit ball there is an action of a discrete subgroup of isometries that fails to have the property of amenability ("Mittelbarkeit” in German). In this talk we will approach the concept of amenability and paradoxcial decompositions from very different perspectives: we will analyze metric, purely algebraic and operator algebraic aspects. In particular, we will define the class of Foelner C*-algebras in terms of a net of unital completely positive maps from the algebra to matrices that are asymptotically multiplicative in a weak sense. This class of C*-algebras include the quasidiagonal ones. Finally, we will present Roe C*-algebras associated to discrete metric spaces with bounded geometry as an example where all these approaches unify. (Joint work with P Ara (UAB), K. Li and J. Wu (U. Münster))."