Riemann was born at Breselenz in Germany and, before studying mathematics in earnest, studied theology in preparation for the priesthood at his father's request. Fortunately he was able to persuade his father, a Lutheran pastor, that his real talents lay elsewhere than in theology. He attended the University of Göttingen and his mathematical abilities were such that his doctoral thesis won the rarely given praise of Karl Friedrich Gauss. After gaining his doctorate Riemann worked on the inaugural lecture necessary in order to gain the post of Privatdozent at Göttingen and this too gained Gauss's praise. Eventually Riemann succeeded his friend, Lejeune Dirichlet, as professor of mathematics at Göttingen in 1859 but by then his health had begun to decline and he died of tuberculosis while on holiday in Italy.

Riemann's work ranges from pure mathematics to mathematical physics and he made influential contributions to both. His first mathematical work was on the functions of a complex variable. His work in analysis was profoundly important. The Riemann integral is a definite integral formally defined in terms of the limit of a summation of elements as the number of elements tends to infinity and their size becomes infinitesimally small.

One of Riemann's most famous pieces of work was in geometry. This was initiated in his inaugural lecture of 1854 that so impressed Gauss, entitled “Concerning the Hypotheses that Underlie Geometry.” What Riemann did was to consider the whole question of what a geometry was from a much more general perspective than anyone had previously done. Riemann asked questions, such as how could concepts like curvature and distance be defined, in such a way as to be applicable to geometries that were not Euclidean. Janós Bolyai and Nikolai Lobachevsky (and, at the time unknown to everyone, Gauss) had developed particular non-Euclidean geometries, but Riemann went further and opened up the possibility of a range of geometries different from Euclid's. This work had far-reaching consequences, not just in pure mathematics but also in the theory of relativity.

Riemann was also interested in applied mathematics and physics and was a coworker of Wilhelm Weber.