The two works titled "What Are and What Should the Numbers Be?" (1888) and "Continuity and Irrational Numbers" (1872) are Dedekinds contributions to the foundations of mathematics; therein, he laid the groundwork for set theory and the theory of real and natural numbers. These writings are indispensable in modern mathematics. However, Dedekinds achievements have not always been adequately acknowledged, and the content of these books is still little known to many mathematicians today. This volume contains not only the original texts but also a detailed analysis of the two writings and an interpretation in modern language, as well as a brief biography and a transcript of the famous letter to H. Keferstein. The extensive commentary offers a fascinating insight into the life and work of Dedekinds pioneering work and relates the latter to great contemporaries such as Cantor, Dirichlet, Frege, Hilbert, Kronecker, and Riemann. Researchers and students alike will find this work a valuable reference in the history of mathematics.