This textbook is written for advanced undegraduates or first year graduate students of mathematics and computer science. It is also intended for the working mathematician who wants to gain an appreciation of Mathematical Logic. There are no prerequisites for this book; however, some mathematical maturity is required. The book is written in a totally mathematical style, and any mathematician should feel at home reading this. The book starts with the definition of first order languages, proceeds through propositional logic, completeness theorems, and finally the two Incompleteness Theorems of Godel. In the process, the reader is also introduced to model theory and recursion Theory. After reading this book, the reader will be ready to branch into model Theory, recursion Theory, axiomatic set theory or even theoretical computer science.Logic is a very profound subject, Godel's work being one the most remarkable discoveries in 20th-Century mathematics, having made an enormous impact on set theory, model theory as well as computer science. In spite of this impact, there exists a dearth of modern books which allow an aspiring mathematician to read and develop an appreciation for the subject. This book attempts to fill that gap by providing a short, distinctive, modern, and well-motivated introduction to mathematical logic.