Based on a course given by the author, which focuses on the inverse problem of Galois Theory: the construction of field extensions having a given finite group as Galois group. This book is devoted to rationality and rigidity criteria and their application in realizing certain groups as Galois groups of regular extensions of Q(T). " is a very stimulating text, which . . . will attract mathematicians working in group theory, number theory, algebraic geometry, and complex analysis. -Zentralblatt für Mathematik This small book contains a nice introduction to some classical highlights and some recent work on the inverse Galois theory problem. The topics and main theorems are carefully chosen and composed in a masterly manner. -Mathematiacl Reviews -July 2007 ""Serre had the great good sense to have notes taken at his 1988 lectures at Harvard, creating a slim volume of great interest..."" -BOOK NEWS Inc., June 2008 J.-P. Serre, one of the greatest mathematicians in our time, provides here a unique introduction to both some classical milestones and some recent developments in the realm of inverse Galois theory. ... [This book] will maintain its unique, unparalleled role in the literature on inverse Galois theory for further generations. Now as before, J.-P. Serre's masterpiece of expository writing is an unvaluable source of inspiration and incitement likewise. -Werner Kleinert, Zentralblatt MATH, January 2007 ""Serre's book helped to call the attention to a deep classical problem with connections to algebraic geometry, topology, algebra, and number theory. By carefully selecting examples, methods and topics, this book goes deeply into the problem."" -MAA Reviews, September 2008"