The third annual CRM Summer School took place in Banff (Alberta, Canada) and was aimed toward advanced students and recent PhDs. This volume presents surveys from the group theory part of the theme year and examines different approaches to the topic: a geometric approach, an approach using methods from logic, and an approach with roots in the BassSerre theory of groups acting on trees. The work offers a concise introduction to current directions of research in combinatorial group theory. Surveys in the text are by leading researchers in the field who are experienced expositors. The text is suitable for use in a graduate course in geometric and combinatorial group theory. Titles in this series are copublished with the Centre de Recherches Mathématiques. Readership Graduate students and research mathematicians working in geometric and combinatorial group theory. Table of Contents  G. Baumslag  Some open problems
 I. M. Chiswell  Length functions and \(\Lambda\)trees
 S. M. Gersten  Introduction to hyperbolic and automatic groups
 O. Kharlampovich and A. Myasnikov  Description of fully residually free groups and irreducible affine varieties over a free group
 Y. Kuz'min  Homological methods in group theory
 M. du Sautoy  Pro\(p\) groups
 S. M. Vovsi  Identities of representations of finite groups
