This book contains the refereed proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Recent Developments in the Inverse Galois Problem, held in July 1993 at the University of Washington, Seattle. A new review of Serre's Topics in Galois Theory serves as a starting point. The book describes the latest research on explicit presentation of the absolute Galois group of the rationals. Containing the first appearance of generalizations of modular curves, the book presents applications that demonstrate the full scope of the Inverse Galois Problem.In particular, the papers collected here show the ubiquity of the applications of the Inverse Galois Problem and its compelling significance. The book will serve as a guide to progress on the Inverse Galois Problem and as an aid in using this work in other areas of mathematics. This includes coding theory and other finite field applications. Group theory and a first course in algebraic curves are sufficient for understanding many papers in the volume. Graduate students will find this an excellent reference to current research, as it contains a list of problems appropriate for thesis material in arithmetic geometry, algebraic number theory, and group theory.

Table of Contents:
Part A. Explicit quotients of $G_{\mathbb Q}$ and $G_{\bar {\mathbb F}(t)}$: Explicit Galois realization of $C_{16}$-extensions of $A_n$ and $S_n$ by T. Crespo Topics in Galois theory by M. D. Fried Parametric solutions of embedding problems by B. H. Matzat Some projective linear groups over finite fields as Galois groups over $\mathbb Q$ by A. Reverter and N. Vila $K$-Admissibility of metacyclic $2$-groups by S. Liedahl and J. Sonn Embedding problems and the $C_{16}\rightarrow C_8$ obstruction by J. R. Swallow Cyclic covers of $\mathbb P^1$ and Galois action on their division points by H. Volklein Part B. Moduli spaces and the structure of $G_{\mathbb Q}$: Introduction to modular towers: Generalizing dihedral group-modular curve connections by M. D. Fried On Galois actions on profinite completions of braid groups by Y. Ihara and M. Matsumoto On the Galois image in the derivation algebra of $\pi_1$ of the projective line minus three points by M. Matsumoto Part C. The structure of $G_{\mathbb R(t)}, G_{\bar {\mathbb F}_q(t)}$, and $G_{\mathbb Q_p(t)}$: Covers of $\mathbb P^1$ over the $p$-adics by P. Debes Existence de points $p$-adiques pour tout $p$ sur un espace de Hurwitz by B. Deschamps Stable models by E. Dew Tout groupe fini est un groupe de Galois sur $\mathbb Q_p(T)$, d'apres Harbater by Q. Liu Specializations of coverings and their Galois groups by W. K. Seiler Rational points and canonical heights on K3-surfaces in $\mathbb P^1\times \mathbb P^1\times \mathbb P^1$ by L. Wang Part D. Group theory and geometric monodromy groups: Mathieu group coverings and linear group coverings by S. S. Abhyankar Fundamental groups for arbitrary categories by P. Feit Monodromy groups of branched coverings: The generic case by R. M. Guralnick and M. G. Neubauer Fundamental groups and embedding problems in characteristic $p$ by D. Harbater On free profinite groups of uncountable rank by M. Jarden Primitive monodromy groups of polynomials by P. Muller.