Eversince, the study of discrete groups and their group rings has attracted researchers from variousmathematical branches and led to beautiful results with proofs involving fields such as number theory,combinatorics and analysis. The basic object of study is the structure of the group G itself, i.e. its subgroups, quotients, etc. and properties of the group ring kG with coefficients in a field k.Recently, techniques such as Randomization and Algebraic Approximation have lead to fruitful insights.This project is focused on new and groundbreaking applications of these two techniques in thestudy of groups and group rings. In order to illustrate this, I am explaining how useful these techniques are by focusing on three interacting topics: (i) new characterizations of amenability related to Dixmier’s Conjecture, (ii) the Atiyah conjecture for discrete groups, and (iii) algebraic approximation in the algebraic K-theory of algebras of functional analytic type. All three problems are presently wide open and progress in any of the three problems would mean a breakthrough in current research.Using Randomization techniques, I want to achieve important results in the understanding of groupsrings by contributing to a better understanding of conjectures of Dixmier’s and Atiyah’s. The field ofAlgebraic Approximation is new, and has already been successfully used by G. Cortinas and myself toresolve a longstanding conjecture in Algebraic K-theory due to Jonathan Rosenberg.