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 equideomposability 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.