Fermat’s equation, which I saw in Carol Gibson’s 1981 edition of Mathematics Dictionary in 2001. I was an average student of Xth Class and was preparing for Mathematical Olympiad. I got very much interested in the equation, as it looked very simple. It took 10 years of study to finally understand the proof of Fermat’s Last Theorem. It comes from Number Theory to The theory of elliptic curves. The mathematicians from 1670 to 1995, devoted themselves to a general proof of FLT. It was an English mathematician, Andrew Wiles who finally found the general proof with help of his doctoral student, R. Taylor. The reviewer considered this proof complete in all respects and was published in Annals of Mathematics, a very prestigious journal. In this book, I have studied the research directions of the problem and possible mathematical analysis is added as supplements. I hope the present edition will help the readers to understand this topic in wider context and thus, leading in the development of the theory of elliptic curves.