Jacobi was born at Potsdam, in Germany. After studying in Berlin, he became a lecturer at Königsberg where he managed to attract the favorable attention of Karl Friedrich Gauss. He was a superb teacher and had an astonishing manipulative skill with formulae. He made a brief but disastrous foray into politics that resulted in his losing a pension he had been granted by the king of Prussia.

Jacobi's most important contributions to mathematics were in the field of elliptic functions. Niels Hendrik Abel had partially anticipated some of Jacobi's work, but the two were equally important in the creation of this subject. Jacobi also worked on Abelian functions and discovered the hyperelliptic functions. He applied his work in elliptic functions to number theory.

Jacobi worked in many other areas of mathematics as well as the theory of functions. He was a pioneer in the study of determinants and a certain type of determinant arising in connection with partial differential equations is known as the Jacobian in his honor. This work was the result of his interest in dynamics, in which field he continued and developed the work of William Hamilton, and produced results that are important in quantum mechanics.