From Pythagoras to Fermat, Euler, and latter-day thinkers, mathematicians have puzzled over the determination of integral solutions of algebraic equations with integral coefficients and with more than one unknown. This text by A. O. Gelfond, an internationally renowned leader in the study of this area, offers a relatively elementary exploration of one of the most challenging problems in number theory. Since equations in integers are encountered in issues related to physics and engineering, the solution of these equations is a matter of practical applications. Nevertheless, the theoretical interest in equations in integers is also worth pursuing because these equations are closely connected with many problems in number theory. This volume's coverage of basic theoretical aspects of such equations promises to widen the horizons of readers from advanced high school students to undergraduate majors in mathematics, physics, and engineering.