This work provides a comprehensive introduction to the non-linear theory of generalized functions (in the sense of Colombeau's construction) on differentiable manifolds. Particular emphasis is laid on a diffeomorphism invariant geometric approach to embedding the space of Schwartz distributions into algebras of generalized functions. The foundations of a "non-linear distributional geometry" are developed, supplying a solid base for an increasing number of applications of algebras of generalized functions to questions of a primarily geometric mature, in particular in mathematical physics. Applications of the resulting theory to symmetry-group analysis of differential equations and the theory of general relativity are presented in separate chapters. These features distinguish the present volume from earlier introductory texts and monographs on the subject.