Minkowski was born at Alexsotas (now in Lithuania but then part of the Russian empire) to parents of Jewish origin. In 1872 the family returned to Germany, settling in Königsberg (now Kaliningrad). Minkowski studied alongside David Hilbert at the University of Königsberg, under Adolf Hurwitz, and gained his PhD in 1885. He taught at Bonn (1885–94) and Königsberg (1894–96) and then worked with Hurwitz at the Zurich Federal Institute of Technology (1896–1902). At Hilbert's instigation a new chair of mathematics was created for Minkowski at the University of Göttingen and he worked there (1902–09) until his death.

In 1883, when still 18, Minkowski was awarded the Grand Prix des Sciences Mathématiques of the Paris Academy of Sciences. The award was shared with Henry J. Smith for their work on the theory of quadratic forms. Minkowski remained occupied with the arithmetic of quadratic forms for the rest of his life. In 1896 he gave a detailed account of his ‘geometry of numbers’ in which he developed geometrical methods for the treatment of certain problems in number theory.

During his short period at Göttingen Minkowski worked closely with David Hilbert and decisively influenced Hilbert's interest in mathematical physics. Minkowski's most celebrated work was in developing the mathematics that played a crucial role in Einstein's formulation of the theory of relativity. Einstein knew when he published the special theory of relativity in 1905 that the universe could not be adequately described using normal, or Euclidean, three-dimensional geometry. Minkowski's seminal idea was to view space and time as forming together a single four-dimensional continuum or manifold, known as space–time, rather than two distinct entities. In normal three-dimensional geometry, any point in space can be identified by three coordinates. The analog of this point in three-dimensional space is an event localized both in space and time in four-dimensional space–time.

Minkowski put forward his concept of space–time, or Minkowski space as it is sometimes called, in 1907 in his book Space and Time. Einstein himself was very forthright about the extent to which the theory of relativity depended on Minkowski's innovatory work. Space–time was a useful and elegant format for special relativity, and was essential for general relativity, published in 1916, in which space–time is allowed to be curved. It is the curvature of space–time that accounts for the phenomenon of gravitation.