This monograph is specially written for students and researchers in mathematics, particularly number theory. Professors and instructors teaching this subject will find it very useful. This monograph develops a clear understanding of the structure of the exponential diophantine 3-term equations and their solutions. Various components of an exponential diophantine equation and their roles are described in detail. A combinatorial approach is used to classify and analyze solutions to exponential diophantine 3-term equations. To classify these solutions, all possible ways these solutions can occur are examined. The analysis presented reveals that these solutions can be classified into three classes and six sub-classes. Properties of each class of solutions are described in detail. We observe that each exponential diophantine 3-term equation imposes its own requirements on its solutions. We discuss how these requirements of an exponential diophantine equation can affect solutions in each class. We apply the classification developed here to relate to the results for the Pythagorean equation, the Catalan equation, and the Ramanujan-Nagell equation. Finally, we employ the concepts developed in this monograph and present twelve proofs and four conjectures that help us analyze and understand the Fermat's Last Theorem and the Beal's conjecture. These four conjectures are much smaller in scope.