T HE WILHELMIAN ERA witnessed an enormous transformation in German mathematics, one that manifested itself not only in new research developments in pure mathematics but also in the emergence of a growing concern for areas where mathematics could potentially be applied. Research in pure mathematics moved away from traditional subjects like elliptic and Abelian functions, algebraic geometry, and invariant theory, fields in which a premium was placed on computational techniques and algorithmic virtuosity. The first decades of the twentieth century saw the proliferation of new research fields like set theory, functional analysis, integral equations, point-set topology, and analytic number theory. Interest in the classical fields of nineteenth-century mathematics often gave way to the study of these new abstract subjects or to generalized axiomatic approaches to the older ones. This fascination with abstract structures crystallized during the Weimar era when Emmy Noether and her Gottingen colleagues launched the new subdiscipline of modern algebra. This led to a movement in twentieth-century mathematics that culminated thirty years later in the monumental work of the French mathematicians who wrote under the collective pseudonym of Nicolas Bourbaki. The line between pure and applied mathematics also began to blur at this time. A substantial part of the new mathematics-matrix algebra, groups and their representations, integral equations, and tensor analysis-proved to be of vital importance for relativity theory and quantum mechanics. Moreover, a good deal of activity was taking place on the applied side of the mathematical spectrum, as new mathematical methods were developed for dealing with a variety of problems posed by construction engineering, ship design, aerodynamics, financial investments, and insurance. The demands of a modern industrialized society were particularly acute in Germany, and mathematicians could ill afford to continue embracing the outmoded neohumanist ideology that had guided their profession throughout the nineteenth century. Although this transformation was obviously a complex process involving the interaction of numerous factors, many of which operated on the macro level, the stage on which much of it was played out was the remote little town of Gottingen. It was there, beginning in 1895, that the two principal actors in this drama, Felix Klein (1849-1925) and David Hilbert (1862-1943), forged an intellectual partnership that not only overturned the balance of power within German mathematics but also significantly altered its research orientation and the relationship between mathematics and other scientific and technological disciplines.