We see that numerous applications to biology, chemistry, economics and medicine have recently been added to the traditional ones in mechanics, hence ordinary differential equations become again a fundamental tool for understanding scientific world. In the last yeares interest for non linear analysis has steadily grown, and the theory of non linear equations supplies a wide field of applications. Therefore the need to write a new book, based on the long experience of the late G. Stampacchia, in which many of the new ideas could be settled together with the classical ones. Thde student that has had a good first course in calculus is thus able to study the book in order to obtain a panoramic vision of the theory. We have tried to use a simple language in order to make the book accessible to engineers and physicists without specific preparation in "modern" mathematics.