In this volume, the authors introduce the theory of existentially closed groups, bringing together both well-established and more contemporary ideas, interpretations, and proofs. They adopt a group-theoretical rather than a model-theoretical point of view as they define existentially closed groups and summarizes some of the techniques that are basic to infinite group theory, such as the formation of free products with amalgamation and HNN-extensions. From this basis the theory is developed and many of the more recently discovered results are proved and discussed.