Stephen Cole Kleene (January 5, 1909, Hartford, Connecticut, United States – January 25, 1994, Madison, Wisconsin) was an American mathematician who helped lay the foundations for theoretical computer science. One of many distinguished students of Alonzo Church, Kleene, along with Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory. Kleene's work grounds the study of which functions are computable. A number of mathematical concepts are named after him: Kleene hierarchy, Kleene algebra, the Kleene star (Kleene closure), Kleene's recursion theorem and the Kleene fixpoint theorem. He also invented regular expressions, and was a leading American advocate of mathematical intuitionism.

Kleene pronounced his last name /ˈkleɪniː/ KLAY-nee; /ˈkliːniː/ and /ˈkliːn/ are common mispronunciations. (His son, Ken Kleene, wrote: "As far as I am aware this pronunciation is incorrect in all known languages. I believe that this novel pronunciation was invented by my father.")

Kleene was awarded the BA degree from Amherst College in 1930. He was awarded the Ph.D. in mathematics from Princeton University in 1934. His thesis, entitled A Theory of Positive Integers in Formal Logic, was supervised by Alonzo Church. In the 1930s, he did important work on Church's lambda calculus. In 1935, he joined the mathematics department at the University of Wisconsin – Madison, where he spent nearly all of his career. After two years as an instructor, he was appointed assistant professor in 1937.

While a visiting scholar at the Institute for Advanced Study in Princeton, 1939 - 40, he laid the foundations for recursion theory, an area that would be his lifelong research interest. In 1941, he returned to Amherst College, where he spent one year as an associate professor of mathematics.

During World War II, Kleene was a lieutenant commander in the United States Navy. He was an instructor of navigation at the U.S. Naval Reserve's Midshipmen's School in New York, and then a project director at the Naval Research Laboratory in Washington, D.C.

In 1946, Kleene returned to Wisconsin, becoming a full professor in 1948 and the Cyrus C. MacDuffee professor of mathematics in 1964. He was chair of the Department of Mathematics and Computer Science, 1962 - 63, and Dean of the College of Letters and Science from 1969 to 1974. The latter appointment he took on despite the considerable student unrest of the day, stemming from the Vietnam War. He retired from the University of Wisconsin in 1979. The mathematics library at the University of Wisconsin was renamed in his honour.

Kleene's teaching at Wisconsin resulted in three texts in mathematical logic (1952, 1967, and with Vesley 1965), often cited and still in print. Kleene (1952) wrote alternative proofs to the Gödel's incompleteness theorems that enhanced their canonical status and made them easier to teach and understand. Kleene and Vesley (1965) is the classic American introduction to intuitionist logic and mathematics.

Kleene served as president of the Association of Symbolic Logic, 1956 - 58, and of the International Union of the History and the Philosophy of Science, 1961. In 1990, he was awarded the National Medal of Science.

Kleene and his spouse Nancy Elliott had four children. He had a lifelong devotion to the family farm in Maine. An avid mountain climber and canoeist, he had a strong interest in nature and the environment and was active in many conservation causes.