Eight expository articles by well-known authors of the theory of Galois groups and fundamental groups focus on recent developments, avoiding classical aspects which have already been described at length in the standard literature. The volume grew from the special semester held at the MSRI in Berkeley in 1999 and many of the new results are due to work accomplished during that program. Among the subjects covered are elliptic surfaces, Grothendieck's anabelian conjecture, fundamental groups of curves and differential Galois theory in positive characteristic. Although the articles contain original results, the authors have striven to make them as introductory as possible, making them accessible to graduate students as well as researchers in algebraic geometry and number theory. The volume also contains a lengthy overview by Leila Schneps that sets the individual articles into the broader context of contemporary research in Galois groups.

### Contents

Introduction; 1. Monodromy groups of coverings of curves Robert Guralnik; 2. On the tame fundamental groups of curves over algebraically closed fields of characteristic > 0 Akio Tamagawa; 3. On the specialization homomorphism of fundamental groups of curves in positive characteristic Florian Pop and Mohamed Saïdi; 4. Topics surrounding the anabelian geometry of hyperbolic curves Shinichi Mochizuki; 5. Monodromy of elliptic surfaces Fedor Bogomolov and Yuri Tschinkel; 6. Tannakian fundamental groups associated to Galois groups Richard Hain and Makoto Matsumoto; 7. Special loci in moduli spaces of curves Leila Schneps; 8. Cellulation of compactified Hurwitz spaces Michel Imbert; 9. Patching and Galois theory David Harbater; 10. Constructive differential Galois theory B. Heinrich Matzat and Marius van der Put.