The main purpose of Tignol’s book is, in his own words, the methodology of mathematics, and the theme used to illustrate the general methodology is the theory of algebraic equations. The main stages of its evolution starting from its origins in Babylonian times to its completion by Galois are reviewed and carefully discussed. Although the statement of the basic problems of the theory of algebraic equations is elementary, requiring only high-school mathematics, its long and eventful development led to profound ideas and to the fundamental concepts of modern algebra. It thus seems to be an ideal topic for the purpose of explaining how mathematics is made: this is why the author relies more on the individual experiences of great mathematicians rather than on the precise development of Galois theory up to modern standards. A pleasant feature is the use of modern notation and terminology to explain the original ideas of Cardano, Viète, Descartes, Newton, Lagrange, Waring, Gauss, Ruffini, Abel, Galois, etc., making the book accessible to any undergraduate student of mathematics, and to any mathematician interested in the historical development of mathematical ideas.