David Hilbert studied at the University of Königsberg under Lindemann. He was friends with Hurwitz, who taught there, and Minkowski, who was also a doctoral student. Both of them were to strongly influence Hilbert's mathematical progress. Hilbert got his doctorate in 1895, and taught at Königsberg for 9 years. In 1895, Hilbert was appointed to the chair of mathematics at the University of Göttingen, where he continued to teach for the rest of his career.

Hilbert's first work was on invariant theory, and in 1888 he proved his famous Basis Theorem. Hilbert submitted a paper on the subject, and despite objections from Gordan, the world expert on invariant theory, it was accepted. He expanded on his methods in a later paper, and Klein, after reading the manuscript, wrote "I do not doubt that this is the most important work on general algebra that the [journal] has ever published."

From 1893 to 1897, Hilbert worked on a book on algebraic number theory. This was a brilliant synthesis of the work of Kummer, Kronecker and Dedekind but contains a wealth of Hilbert's own ideas. The beginnings of Class field theory are all contained in this work.

Hilbert's work in geometry had the greatest influence in that area since Euclid. A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analysed their significance. He published a paper in 1899 putting geometry in a formal axiomatic setting. The book continued to appear in new editions and was a major influence in promoting the axiomatic approach to mathematics during the twentieth century.

Hilbert gave a famous speech, "The Problems of Mathematics", to the Second International Congress of Mathematicians in Paris. In it he presented 23 unsolved problems he felt were fundamental to mathematics. Hilbert's problems included the continuum hypothesis, the well ordering of the reals, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet's principle and many more. Many of the problems were solved during the twentieth century, and each time one of the problems was solved it was a major event for mathematics.

Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations. Hilbert's work in integral equations began the field of functional analysis. This work also established the basis for his work on infinite-dimensional space, later called Hilbert space. Making use of his results on integral equations, Hilbert contributed to the development of mathematical physics by his important memoirs on kinetic gas theory and the theory of radiations.

Among Hilbert's students were Hermann Weyl, the famous world chess champion Lasker, and Zermelo. His first student, Otto Blumenthal, wrote "Insofar as the creation of new ideas is concerned, I would place [others] higher, but when it comes to penetrating insight, only a few of the very greatest were the equal of Hilbert".

Hilbert received many honors. In 1905 the Hungarian Academy of Sciences gave a special citation for Hilbert. In 1930 Hilbert retired and the city of Königsberg made him an honorary citizen of the city.