Two main objects of the research in this thesis are countable discrete groups and their operator algebras (C*-algebras and von Neumann algebras). Discrete groups are often succesfully studied using geometric and ergodic-theoretic methods, the corresponding areas of mathematics being called geometric resp. measured group theory. This thesis has a cumulative form: each chapter is a research article, and therefore has its own abstract and bibliography. The majority of these publications have been peer-reviewed and published in various journals.