Although this is an introductory text on proof theory, most of its contents has not previously been presented in a unified form in the literature, except at a very advanced level. The heart of the book is the ordinal analysis of axiom systems, with particular emphasis on that of the impredicative theory of elementary inductive definitions on the natural numbers. The "constructive" consequences of ordinal analysis are sketched out in the epilogue. The book sets out to give a self-contained treatment assuming no prior knowledge of proof theory and almost none of logic. The author has, moreover, been at pains not to use the "cabal language" of proof theory, but only a language familiar to most readers.