The fundamental property of compact spacesthat continuous functions defined on compact spaces are boundedserved as a motivation for E. Hewitt to introduce the notion of a pseudocompact space. The class of pseudocompact spaces proved to be of fundamental importance in settheoretic topology and its applications. This clear and selfcontained exposition offers a comprehensive treatment of the question, When does a group admit an introduction of a pseudocompact Hausdorff topology that makes group operations continuous? Equivalently, what is the algebraic structure of a pseudocompact Hausdorff group? The authors have adopted a unifying approach that covers all known results and leads to new ones. Results in the book are free of any additional settheoretic assumptions. Readership Graduate students and research mathematicians working in algebra, set theory and topology. Table of Contents  Introduction
 Principal results
 Preliminaries
 Some algebraic and settheoretic properties of pseudocompact groups
 Three technical lemmas
 Pseudocompact group topologies on \(\mathcal V\)free groups
 Pseudocompact topologies on torsion Abelian groups
 Pseudocompact connected group topologies on Abelian groups
 Pseudocompact topologizations versus compact ones
 Some diagrams and open questions
 Diagram 2
 Diagram 3
 Bibliography
