Born in Liegnitz, now Legnica in Poland, Kronecker studied mathematics at Berlin but he did not become a professional mathematician until relatively late in life. He worked, highly successfully, as a businessman until he had made enough money to abandon commerce and devote himself fully to mathematics. He taught at Berlin from 1861, and, in 1883, was appointed professor. Outside mathematics Kronecker's interests were wide. He was a highly cultured man who used his wealth to patronize the arts. He also had a deep interest in philosophy and Christian theology, although he was not converted to Christianity until shortly before his death.

Kronecker's mathematical work was almost entirely in the fields of number theory and higher algebra, although he also made some contributions to the theory of elliptic functions. His work on algebraic numbers was inspired by his constructivist outlook, which involved a distrust of nonconstructive proofs in mathematics and a suspicion of the infinite and all kinds of number other than the natural numbers. This attitude led him to rewrite large areas of algebraic number theory in order to avoid reference to such suspect entities as imaginary or irrational numbers. Kronecker's constructivism is summed up in a famous remark he made during an after-dinner speech: “God made the integers, all else is the work of man.” His suspicion of nonconstructive methods led Kronecker into fierce controversy with two of the leading mathematicians of his day, Karl Weierstrass and Georg Cantor. His outlook anticipates to a considerable extent the views of the Dutch mathematician L. E. J. Brouwer.

Kronecker was also one of the first to understand thoroughly and use Evariste Galois's work in the theory of equations. The Kronecker delta function is named for him.