This book contains the latest developments of the theory of discontinuous groups acting on homogenous spaces, from basic concepts to a comprehensive exposition. It develops the newest approaches and methods in the deformation theory of topological modules and unitary representations and focuses on the geometry of discontinuous groups of solvable Lie groups and their compact extensions. It also presents proofs of recent results, computes fundamental examples, and serves as an introduction and reference for students and experienced researchers in Lie theory, discontinuous groups, and deformation (and moduli) spaces.