This book contains a reprint of thirteen lectures by Cauchy on ordinary differential equations. This printed version, whose existence was long doubted, was discovered by the editor, M. Christian Gilain in the library of the Institute. The first four lectures contain standard information on the cloud form solution of equations of standard type. The last seven lectures contain Cauchy’s investigation into existence, uniqueness, and approximation by power series. As Jean Dieudonné remarks in his preface, these lectures show that Cauchy had discovered the essence of the Cauchy-Lipschitz method by 1820 or 1821 at the latest. The book contains, in addition to Dieudonné’s preface in French and English, a long essay by the editor detailing the history of the lectures and an analysis of Cauchy’s work on differential equations.