This book verifies with compelling evidence the author’s inclination to ''write a book on proof theory which needs no previous knowledge of proof theory''. Avoiding the cryptic terminology of proof as far as possible, the book starts at an elementary level and displays the connections between infinitary proof theory and generalized recursion theory, especially the theory of inductive definitions. As a ''warm up'' the classical analysis of Gentzen is presented in a more modern terminology to proceed with explaining and proving the famous result by Feferman and Schütte on the limits of predicativity. The author, too, provides an introduction to ordinal arithmetic, introduces the Veblen hierarchy and employs these functions to design an ordinal notation system for the ordinals below Epsilon 0 and Gamma 0, while emphasizing the first step into impredicativity, i.e., the first step beyond Gamma 0.An earlier version of this book was originally published in 1989 as volume 1407 of the Springer series ''Lecture Notes in Mathematics''.