The present book is the first to systematically treat the theory of groups generated by a conjugacy class of subgroups, satisfying certain generational properties on pairs of subgroups. For finite groups, this theory has been developed in the 1970s mainly by M. Aschbacher, B. Fischer and the author. It was extended to arbitrary groups in the 1990s by the author. The theory of abstract root subgroups is an important tool to study and classify simple classical and Lie-type groups. It is strongly related to the theory of root groups on buildings developed by J. Tits, which in turn extends the theory of root subgroups of Chevalley groups. The book is of interest to mathematicians working in different areas such as finite group theory, classsical groups, algebraic and Lie-type groups, buildings and generalized polygons. It will also be welcomed by the graduate student in any of the above subjects, as well as the researcher working in any of these areas. Parts of it can also be used for graduate classes. Large parts of the book are self-contained and accessible with reasonable knowledge in abstract group theory and classical groups. Its main purpose is to give complete and partially new proofs of results that are quite unaccessible in the literature.