Part I: General Existence of Solutions Theory. 1. Introduction. 2. Approximation of Solutions of Continuous Nonlinear PDEs. 3. Spaces of Generalized Functions. 4. Extending T(x,D) to the Order Completion of Spaces of Smooth Functions. 5. Existence of Generalized Solutions. 6. A Few First Examples. 7. Generalized Solutions as Measurable Functions. Part II: Applications to Specific Classes of Linear and Nonlinear PDEs. 8. The Cauchy Problem for Nonlinear First Order Systems. 9. An Abstract Existence Result. 10. PDEs with Sufficiently Many Smooth Solutions. 11. Nonlinear Systems with Measures as Initial Data. 12. Solution of PDEs and the Completion of Uniform Spaces. 13. Partial Orders Compatible with a Nonlinear Partial Differential Operator. 14. Miscellaneous Results. Part III: Group Invariance of Global Generalized Solutions of Nonlinear PDEs. 15. Introduction. 16. Group Invariance of Global Generalized Solutions of Nonlinear PDEs Obtained Through the Algebraic Method. 17. Group Invariance of Generalized Solutions Obtained Through the Algebraic Method: An Alternative Approach. 18. Group Invariance of Global Generalized Solutions Obtained Through the Order Completion Method. Appendix. References. Index.