Introduction to ℓ² invariants This book introduces the reader to the most important concepts and problems in the field of ℓ² invariants. After some foundational material on group von Neumann algebras ℓ² Betti numbers are defined and their use is illustrated by several examples. The text continues with Atiyahs question on possible values of ℓ² Betti numbers and the relation to Kaplanskys zero divisor conjecture. The general definition of ℓ² Betti numbers allows for applications in group theory. A whole chapter is dedicated to Lücks approximation theorem and its generalizations. The final chapter deals with ℓ² 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