This book sets forth the general theory of Lie and his contemporaries and the results of recent investigations with the aid of the methods of the tensor calculus and concepts of the new differential geometry. The first three chapters contain in the main the results of the first period. Chapter Four is devoted to the theory of the adjoint group and the sequence of theorems basic to the characterization of semi-simple groups, as developed by Cartan and recently by Weyl and Schouten. Although geometrical ideas are used throughout the book, Chapter Five contains the geometrical applications of the theory both in the space of the transformations and in the group-space, and here are to be found particularly the concepts of the new differential geometry. The theory of contact transformations with applications to geometry and mechanics is set forth in the closing chapter.