Hilbert and Bernays' 'Grundlagen der Mathematik'appeared in two volumes in 1934 and 1939, asecond edition in 1968 and 1970. It offers afoundation for proof theory and is a major source onDavid 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 groundand philosophical perspectives beyond the scope ofWhitehead and Russell's 'Principia Mathematica' andFrege's 'Grundlagen der Arithmetik'.This book is not only essential to any scholar of thehistory and philosophy of modern mathematics, but italso contains formal research - on the epsilon and iotaoperators - of contemporary relevance to logicians, mathematicians and computer science. For us, it isone of the most fascinating books ever written.This a bilingual German-English, commented editionof the 'Grundlagen'. It is the fi rst English publicationof these texts and shows the facsimile of the Germanoriginal 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 historyand the interpretation of the text's mathematicaland philosophical content, there are also carefulannotations regarding the differences between thetwo German editions (1934/39,1968/1970) of thistwo volume monograph.