Describing the geometric theory of discrete groups and the associated tesselations of the underlying space, this work also develops the theory of Mobius transformations in n-dimensional Euclidean space. These transformations are discussed as isometries of hyperbolic space and are then identified with the elementary transformations of complex analysis. A detailed account of analytic hyperbolic trigonometry is given, and this forms the basis of the subsequent analysis of tesselations of the hyperbolic plane. Emphasis is placed on the geometrical aspects of the subject and on the universal constraints which must be satisfied by all tesselations.