The Denjoy integral has not found many applications in analysis. Such applications as have been made are due mainly to authors of this book. The original papers are in difficult to obtain Soviet journals. This translation makes this important work available to wider audience and shows for the first time in English that the Denjoy integral has applications in both real and complex analysis. In addition, the longest chapter of the book is devoted to the theory of a two dimensional extension of the wide Denjoy integral. The book is a self contained treatment of the integral in one and two dimensions, together with details of most known applications of these integrals to analysis, both real and complex.