One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory, which subsequently helped to provide the foundations of theoretical computer science.

He also invented regular expressions in 1951 to describe McCulloch-Pitts neural networks, and made significant contributions to the foundations of mathematical intuitionism.

He was awarded a Ph.D. in mathematics from Princeton University in 1934, where his thesis, entitled A Theory of Positive Integers in Formal Logic, was supervised by Alonzo Church.

While a visiting scholar at the Institute for Advanced Study in Princeton, 1939–1940, he laid the foundation for recursion theory, an area that would be his lifelong research interest.

Kleene (1952) wrote alternative proofs to the Gödel's incompleteness theorems that enhanced their canonical status and made them easier to teach and understand.

In the summer of 1951 at the Rand Corporation, he produced a major breakthrough in a third area when he gave an important characterization of events accepted by a finite automaton.

[4]Kleene served as president of the Association for Symbolic Logic, 1956–1958, and of the International Union of History and Philosophy of Science,[5] 1961.