Alexander S. Kechris is Professor of Mathematics at the California Institute of Technology. He is the recipient of numerous honors, including the Sloan Research Fellowship, the J. S. Guggenheim Memorial Foundation Fellowship, and the Carol Karp Prize of the Association for Symbolic Logic. He is also an Inaugural Fellow of the American Mathematical Society.

• 1. Equivalence relations and reductions; 2. Countable Borel equivalence relations; 3. Essentially countable relations; 4. Invariant and quasi-invariant measures; 5. Smoothness, $\mathbf{E}_0$ and $\mathbf{E}_\infty$; 6. Rigidity and incomparability; 7. Hyperfiniteness; 8. Amenability; 9. Treeability; 10. Freeness; 11. Universality; 12. The poset of bireducibility types; 13. Structurability; 14. Topological realizations; 15. A universal space for actions and equivalence relations; 16. Open problems; References; List of Notation; Subject Index.

Alexander S. Kechris is Professor of Mathematics at the California Institute of Technology. He is the recipient of numerous honors, including the Sloan Research Fellowship, the J. S. Guggenheim Memorial Foundation Fellowship, and the Carol Karp Prize of the Association for Symbolic Logic. He is also an Inaugural Fellow of the American Mathematical Society.

• 1. Equivalence relations and reductions; 2. Countable Borel equivalence relations; 3. Essentially countable relations; 4. Invariant and quasi-invariant measures; 5. Smoothness, $\mathbf{E}_0$ and $\mathbf{E}_\infty$; 6. Rigidity and incomparability; 7. Hyperfiniteness; 8. Amenability; 9. Treeability; 10. Freeness; 11. Universality; 12. The poset of bireducibility types; 13. Structurability; 14. Topological realizations; 15. A universal space for actions and equivalence relations; 16. Open problems; References; List of Notation; Subject Index.