The main theme of this book is a study of properties of a certain class of infinite-dimensional groups. It is very much an interdisciplinary topic having relations to many parts of mathematics, including representations theory, descriptive set theory, infinite dimensional Lie groups, topological transformation groups, operator algebras, topological and ergodic dynamics and Ramsey theory. An approach presented in the book describes relations among dynamical properties of these groups, asymptotic geometric analysis, Ramsey type theorems in combinatorics and concentration properties. A presentation of a circle of problems relating the mentioned topics is based on a detailed study of representative examples (e.g. the unitary group of a Hilbert space, the group of homeomorphisms of a manifold, the infinite symmetric group, the group of transformations of measure spaces) and it covers many recent results in the field.