Hilbert studied at the university in his native city of Königsberg (now Kaliningrad in Russia) and at Heidelberg; he also spent brief periods in Paris and Leipzig. He took his PhD in 1885, the next year became Privatdozent at Königsberg, and by 1892 had become professor there. In 1895 he moved to Göttingen to take up the chair that he occupied until his official retirement in 1930.

Hilbert's mathematical work was very wide ranging and during his long life there were few fields to which he did not make some contribution and many he completely transformed. His attention was first turned to the newly created theory of invariants and in the period 1885–88 he virtually completed the subject by solving all the central problems. However his work on invariants was very fruitful as he created entirely new methods for tackling problems, in the context of a much wider general theory. The fruit of this work consisted of many new and fundamental theorems in algebra and in particular in the theory of polynomial rings. Much of his work on invariants turned out later to have important application in the new subject of homological algebra.

Hilbert now turned to algebraic number theory where he did what is probably his finest research. Hilbert and Minkowski had been asked to prepare a report surveying the current state of number theory but Minkowski soon dropped out leaving Hilbert to produce not only a masterly account but also a substantial body of original and fundamental new discoveries. The work was presented in the Zahlbericht (1897; Report on Numbers) with an elegance and lucidity of exposition that has rarely been equalled.

Hilbert then moved to another area of mathematics and wrote the Grundlagen der Geometrie (1899; Foundations of Geometry), giving an account of geometry as it had developed through the 19th century. Here his interest lay chiefly in expounding and illuminating the work of others in a systematic way rather than in making new developments of the subject. He devised an abstract axiomatic system that could admit many different geometries – Euclidean and non-Euclidean – as models and by this means go much further than had previously been done in obtaining consistency and independence proofs for various sets of geometrical axioms. Apart from its importance for pure geometry his work led to the development of a number of new algebraic concepts and was particularly important to Hilbert himself because his experience with the axiomatic method and his interest in consistency proofs shaped his approach to mathematical logic and the foundations of mathematics.

In mathematical logic and the philosophy of mathematics Hilbert is a key figure, being one of the major proponents of the formalist view, which he expounded with much greater precision than had his 19th-century precursors. This philosophical view of mathematics had a formative impact on the development of mathematical logic because of the central role it gave to the formalization of mathematics into axiomatic systems and the study of their properties by metamathematical means. Hilbert aimed at formalizing as much of mathematics as possible and finding consistency proofs for the resulting formal systems. It was soon shown by Kurt Gödel that Hilbert's program, as this proposal is called, could not be carried out, at least in its original form, but it is none the less true that Gödel's own revolutionary metamathematical work would have been inconceivable without Hilbert. Hilbert's contribution to mathematical logic was important, especially to the development of proof theory, as further developed by such mathematicians as Gerhard Gentzen.

Hilbert also made notable contributions to analysis, to the calculus of variations, and to mathematical physics. His work on operators and on Hilbert space (a type of infinite-dimensional space) was of crucial importance to quantum mechanics. His considerable influence on mathematical physics was also exerted through his colleagues at Göttingen, who included Minkowski, Hermann Weyl, Erwin Schrödinger, and Werner Heisenberg.

In 1900 Hilbert presented a list of 23 outstanding unsolved mathematical problems to the International Congress of Mathematicians in Paris. A number of these problems still remain unsolved and the mathematics that has been created in solving the others has fully vindicated his deep insight into his subject. Hilbert was an excellent teacher and during his time at Göttingen continued the tradition begun in the 19th century and built the university into an outstanding center of mathematical research, which it remained until the dispersal of the intellectual community by the Nazis in 1933. Hilbert is generally considered one of the greatest mathematicians of the 20th century and indeed of all time.