Important results surrounding the proof of Goldbachs ternary conjecture are presented in this book. Beginning with an historical perspective along with an overview of essential lemmas and theorems, this monograph moves on to a detailed proof of Vinogradovs theorem. The principles of the Hardy-Littlewood circle method are outlined and applied to Goldbachs ternary conjecture. New results due to H. Maier and the author on Vinogradovs theorem are proved under the assumption of the Riemann hypothesis. The final chapter discusses an approach to Goldbachs conjecture through theorems by L. G. Schnirelmann. This book concludes with an Appendix featuring a sketch of H. Helfgotts proof of Goldbachs ternary conjecture. The Appendix also presents some biographical remarks of mathematicians whose research has played a seminal role on the Goldbach ternary problem.  The authors step-by-step approach makes this book accessible to those that have mastered classical number theory and fundamental notions of mathematical analysis. This book will be particularly useful to graduate students and mathematicians in analytic number theory, approximation theory as well as to researchers working on Goldbachs problem.