The present volume is the fifth in a series of six, presenting a selection of the previously unpublished lecture notes of David Hilbert on the foundations of mathematics and natural science, roughly spanning the period from 1890 to 1933. Hilbert's Lectures and his personal interactions with the 'Hilbert circle' exercised a profound influence on the development of twentieth century mathematics and physics. The lecture notes presented, spanning virtually the whole of Hilbert's teaching career, document his intense engagement with the ideas of some of the central figures of modern science and make possible a detailed understanding of the development of his foundational work in geometry, arithmetic, logic, and proof theory, as well as in the theory of relativity, quantum mechanics and statistical physics. The lectures contain more philosophical, foundational and methodological remarks than does Hilbert's published work. Some of the individual volumes also reprint key published works of Hilbert when these are centrally relevant to the unpublished work presented. – Volume 5 has three parts, dealing with General Relativity, Epistemological Issues, and Quantum Mechanics. The core of the first part is Hilbert’s two semester lecture course on ‘The Foundations of Physics’ (1916/17). This is framed by Hilbert’s published ‘First and Second Communications’ on the ‘Foundations of Physics’ (1915, 1917) and by a selection of documents dealing with more specific topics like ‘The Principle of Causality’ or a lecture on the new concepts of space and time held in Bucharest in 1918. The epistemological issues concern the intricate relation between nature and mathematical knowledge, in particular the question of irreversibility and objectivity (1921) as well as the subtle question whether what Hilbert calls the ‘world equations’ are physically complete (1923). The last part deals with quantum theory in its early, advanced and mature stages. Hilbert held lecture courses on the mathematical foundations of quantum theory twice, before and after the breakthrough in 1926. These documents bear witness to one of the most dramatic changes in the foundations of science. M.-M. V.