Intensive study of the theory and geometrical applications of continuous groups of transformations provides extended discussions of tensor analysis, Riemannian geometry and its generalizations, and the applications of the theory of continuous groups to modern physics. Contents: 1. The Fundamental Theorems. 2. Properties of Groups. Differential Equations. 3. Invariant Sub-Groups. 4. The Adjoint Group. 5. Geometrical Properties. 6. Contact Transformations. Bibliography. Index. Unabridged republication of the 1933 first edition.