This book presents a large amount of material, both classic and recent (on occasion, unpublished) about the relations of Algebra and Topology. It therefore belongs to the area called Topological Algebra. More specifically, the objects of the study are subtle and sometimes unexpected phenomena that occur when the continuity meets and properly feeds an algebraic operation. Such a combination gives rise to many classic structures, including topological groups and semigroups, paratopological groups, etc. Special emphasis is given to tracing the influence of compactness and its generalizations on the properties of an algebraic operation, causing on occasion the automatic continuity of the operation. The main scope of the book, however, is outside of the locally compact structures, thus distinguishing the monograph from a series of more traditional textbooks. The book is unique in that it presents very important material, dispersed in hundreds of research articles, not covered by any monograph in existence. The reader is gently introduced to an amazing world at the interface of Algebra, Topology, and Set Theory. He/she will find that the way to the frontier of the knowledge is quite short — almost every section of the book contains several intriguing open problems whose solutions can contribute significantly to the area. Contents: Introduction to Topological Groups and Semigroups; Right Topological and Semitopological Groups; Topological Groups: Basic Constructions; Some Special Classes of Topological Groups; Cardinal Invariants of Topological Groups; Moscow Topological Groups and Completions of Groups; Free Topological Groups; R-Factorizable Topological Groups; Compactness and its Generalizations in Topological Groups; Actions of Topological Groups on Topological Spaces.