This book introduces the reader to the most important concepts and problems in the field of -invariants. After some foundational material on group von Neumann algebras, -Betti numbers are defined and their use is illustrated by several examples. The text continues with Atiyah's question on possible values of -Betti numbers and the relation to Kaplansky's zero divisor conjecture. The general definition of -Betti numbers allows for applications in group theory. A whole chapter is dedicated to Lck's approximation theorem and its generalizations. The final chapter deals with -torsion, twisted variants and the conjectures relating them to torsion growth in homology.The text provides a self-contained treatment that constructs the required specialized concepts from scratch. It comes with numerous exercises and examples, so that both graduate students and researchers will find it useful for self-study or as a basis for an advanced lecture course.graduate students and researchers will find it useful for self-studying or as basis for an advanced lecture course.