The first part of the book presents the elementary facts of algebraic geometry essential to understanding the rest of it. The second half of the book considers the evolution of the theory of Diophantine equations from the Renaissance to the middle of the 20th century. In particular, the book includes substantial descriptions of the relevant contributions of Viиte, Fermat, Euler, Jacobi, and Poincarй. The book ends with Joseph Silverman's survey of Diophantine analysis during the last 20 years in which he mentions the proof of the Mordei conjecture and of Fermat's Last Theorem.