Hermann Minkowski was born in Aleksotas, a village in the Kovno Governorate of the Russian Empire (now incorporated into the city of Kaunas, Lithuania) to Lewin Boruch Minkowski, a merchant who subsidized the building of the choral synagogue in Kovno, and Rachel Taubmann, both of Jewish descent. Hermann was a younger brother of the medical researcher, Oskar (born 1858). In different sources Minkowski's nationality is variously given as German, Polish, or Lithuanian-German, or Russian.

To escape persecution in Russia the family moved to Königsberg in 1872, where the father became involved in rag export and later in manufacture of mechanical clockwork tin toys (he operated his firm Lewin Minkowski & Son with his eldest son Max).

Minkowski was educated in East Prussia at the Albertina University of Königsberg, where he earned his doctorate in 1885 under the direction of Ferdinand von Lindemann. In 1883, while still a student at Königsberg, he was awarded the Mathematics Prize of the French Academy of Sciences for his manuscript on the theory of quadratic forms. He also became a friend of another renowned Mathematician, David Hilbert. His brother, Oskar Minkowski (1858–1931), was a well-known physician and researcher.

Minkowski studied in Königsberg and taught in Bonn (1887–1894), Königsberg (1894–1896) and Zurich (1896–1902), and finally in Göttingen from 1902 until his premature death in 1909. He married Auguste Adler in 1897 with whom he had two daughters; the electrical Engineer and Inventor Reinhold Rudenberg was his son-in-law.

Minkowski explored the arithmetic of quadratic forms, especially concerning n variables, and his research into that topic led him to consider certain geometric properties in a space of n dimensions. In 1896, he presented his geometry of numbers, a geometrical method that solved problems in number theory. He is also the creator of the Minkowski Sausage and the Minkowski cover of a curve.

In 1902, he joined the Mathematics Department of Göttingen and became a close colleague of David Hilbert, whom he first met at university in Königsberg. Constantin Carathéodory was one of his students there.

Minkowski is perhaps best known for his work in relativity, in which he showed in 1907 that his former student Albert Einstein's special theory of relativity (1905) could be understood geometrically as a theory of four-dimensional space–time, since known as the "Minkowski spacetime".

By 1907 Minkowski realized that the special theory of relativity, introduced by his former student Albert Einstein in 1905 and based on the previous work of Lorentz and Poincaré, could best be understood in a four-dimensional space, since known as the "Minkowski spacetime", in which time and space are not separated entities but intermingled in a four dimensional space–time, and in which the Lorentz geometry of special relativity can be effectively represented. The beginning part of his address delivered at the 80th Assembly of German Natural Scientists and Physicians (21 September 1908) is now famous:

Minkowski died suddenly of appendicitis in Göttingen on 12 January 1909. David Hilbert's obituary of Minkowski illustrates the deep friendship between the two mathematicians (translated):

Notice of Minkowski's death was communicated to the Quaternion Society in 1910 by its President, Alexander Macfarlane, who had explored hyperbolic quaternions as the "Algebra of Space":

Einstein at first viewed Minkowski's treatment as a mere mathematical trick, before eventually realizing that a geometrical view of space–time would be necessary in order to complete his own later work in general relativity (1915).