This project is a second edition of the textbook: Deformation Theory of Discontinuous Groups (De Gruyter 2022). It is devoted to studying various geometric and topological concepts related to the deformation and moduli spaces of discontinuous group actions, and building some interrelationships between these concepts. It presents full proofs of recent results, computes fundamental examples, and serves as an introduction and reference for students and researchers in Lie theory, discontinuous groups, and deformation spaces. A part of the first edition, the setting of affine actions is introduced and new ideas and methods are developed with full proofs. The setting of compact extensions is also re-written with new approaches and proofs. It also contains the most recent developments of the theory, extending from basic concepts to a comprehensive exposition, and highlighting the newest approaches and methods in deformation theory. It also includes the most recent solutions to many open questions over the last decades and brings related newest research results in this area. For specialists and beginners in deformation theory, the settings of Heisenberg and Threadlike cases are differently re-written with full details and proofs.
