This text is an exposition of what is known about the fundamental groups of compact Kahler manifolds. This class of groups contains all finite groups and is strictly smaller than the class of all finitely presentable groups. This book collects together results obtained in recent years which aim to characterize those infinite groups which can arise as fundamental groups of compact Kahler manifolds. Most of these results are negative ones, saying which groups do not arise. The methods and techniques used form a mix of topology, differential and algebraic geometry, and complex analysis. The book...