Renzo Cavalieri is Associate Professor of Mathematics at Colorado State University. He received his PhD in 2005 at the University of Utah under the direction of Aaron Bertram. Hurwitz theory has been an important feature and tool in Cavalieri's research, which revolves around the interaction among moduli spaces of curves and maps from curves, and their different compactifications. He has taught courses on Hurwitz theory at the graduate and undergraduate level at Colorado State University and around the world at the National Institute for Pure and Applied Mathematics (IMPA) in Brazil, Beijing University, and the University of Costa Rica. Eric Miles is Assistant Professor of Mathematics at Colorado Mesa University. He received his PhD in 2014 under the supervision of Renzo Cavalieri. Miles' doctoral work was on Bridgeland Stability Conditions, an area of algebraic geometry that makes significant use of homological algebra.

'To wit, the book is indeed well-suited to advanced undergraduates who know some serious algebra, analysis (complex analysis in particular), and are disposed to hit themes in algebraic topology and (to a limited degree) algebraic geometry. It would make a good text for a senior seminar.' Michael Berg, MAA Reviews

• Introduction; 1. From complex analysis to Riemann surfaces; 2. Introduction to manifolds; 3. Riemann surfaces; 4. Maps of Riemann surfaces; 5. Loops and lifts; 6. Counting maps; 7. Counting monodromy representations; 8. Representation theory of Sd; 9. Hurwitz numbers and Z(Sd); 10. The Hurwitz potential; Appendix A. Hurwitz theory in positive characteristic; Appendix B. Tropical Hurwitz numbers; Appendix C. Hurwitz spaces; Appendix D. Does physics have anything to say about Hurwitz numbers?; References; Index.

Renzo Cavalieri is Associate Professor of Mathematics at Colorado State University. He received his PhD in 2005 at the University of Utah under the direction of Aaron Bertram. Hurwitz theory has been an important feature and tool in Cavalieri's research, which revolves around the interaction among moduli spaces of curves and maps from curves, and their different compactifications. He has taught courses on Hurwitz theory at the graduate and undergraduate level at Colorado State University and around the world at the National Institute for Pure and Applied Mathematics (IMPA) in Brazil, Beijing University, and the University of Costa Rica. Eric Miles is Assistant Professor of Mathematics at Colorado Mesa University. He received his PhD in 2014 under the supervision of Renzo Cavalieri. Miles' doctoral work was on Bridgeland Stability Conditions, an area of algebraic geometry that makes significant use of homological algebra.

'To wit, the book is indeed well-suited to advanced undergraduates who know some serious algebra, analysis (complex analysis in particular), and are disposed to hit themes in algebraic topology and (to a limited degree) algebraic geometry. It would make a good text for a senior seminar.' Michael Berg, MAA Reviews

• Introduction; 1. From complex analysis to Riemann surfaces; 2. Introduction to manifolds; 3. Riemann surfaces; 4. Maps of Riemann surfaces; 5. Loops and lifts; 6. Counting maps; 7. Counting monodromy representations; 8. Representation theory of Sd; 9. Hurwitz numbers and Z(Sd); 10. The Hurwitz potential; Appendix A. Hurwitz theory in positive characteristic; Appendix B. Tropical Hurwitz numbers; Appendix C. Hurwitz spaces; Appendix D. Does physics have anything to say about Hurwitz numbers?; References; Index.