Hilbert and Bernays' 'Grundlagen der Mathematik' appeared in two volumes in 1934 and 1939, a second edition in 1968 and 1970. It offers a foundation for proof theory and is a major source on David Hilbert's formalist programme, Paul Bernays' philosophy, the epsilon operator, and much more. It has been a profound infl uence on mathematics, logic, and philosophy, and it covers formal ground and philosophical perspectives beyond the scope of Whitehead and Russell's 'Principia Mathematica' and Frege's 'Grundlagen der Arithmetik'. This book is not only essential to any scholar of the history and philosophy of modern mathematics, but it also contains formal research - on the epsilon and iota operators - of contemporary relevance to logicians, mathematicians and computer science. For us, it is one of the most fascinating books ever written. This a bilingual German-English, commented edition of the 'Grundlagen'. It is the fi rst English publication of these texts and shows the facsimile of the German original text on the left-hand side of a double page, and its English translation on the right-hand side. In addition to extensive comments on the history and the interpretation of the text's mathematical and philosophical content, there are also careful annotations regarding the differences between the two German editions (1934/39,1968/1970) of this two volume monograph.