An introduction to harmonic measure on plane domains and careful discussion of the work of Makarov, Carleson, Jones and others.

John B. Garnett is Professor of Mathematics in the Department of Mathematics at University of California, Los Angeles.

1. Jordan domains; 2. Finitely connected domains; 3. Potential theory; 4. Extremal distance; 5. Applications and reverse inequalities; 6. Simply connected domains, part one; 7. Bloch functions and quasicircles; 8. Simply connected domains, part two; 9. Infinitely connected domains; 10. Rectifiability and quadratic expressions; Appendices.