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Part I. Paradoxical Decompositions, or the Nonexistence of Finitely Additive Measures: 1. Introduction 2. The Hausdorff paradox 3. The Banach-Tarski paradox: duplication spheres and balls 4. Locally commutative actions: minimizing the number of pieces in a paradoxical decomposition 5. Higher dimensions and non-Euclidean spaces 6. Free groups of large rank: getting a continuum of spheres from one 7. Paradoxes in low dimensions 8. The semi-group of equidemomposability types Part II. Finitely Additive Measures, or the Nonexistence of Paradoxical Decompositions: 9. Transition 10. Measures in groups 11. Applications of amenability: Marczewski measures and exotic measures 12. Growth conditions in groups and supramenability 13. The role of the axiom of choice.