Kleene was born in Hartford, Connecticut, USA. He received his bachelor of arts degree from Amherst College in 1930. From 1930 to 1935, he was a graduate student and research assistant at Princeton University, where he received his doctorate in mathematics in 1934, supervised by Alonzo Church, for a thesis entitled A Theory of Positive Integers in Formal Logic. In 1935, he joined the UW-Madison mathematics department as an instructor. He became an assistant professor in 1937.
From 1939 to 1940, he was a visiting scholar at Princeton's Institute for Advanced Study, where he laid the foundation for recursive function theory, an area that would be his lifelong research interest. In 1941 he returned to Amherst 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, he returned to Wisconsin, becoming a full professor in 1948. He was chair of mathematics and computer sciences in 1962 and 1963, and dean of the College of Letters and Science from 1969 to 1974. In 1964 he was named the Cyrus C. MacDuffee professor of mathematics. He retired in 1979.
An avid mountain climber, Kleene had a strong interest in nature and the environment and was active in many conservation causes. He led several professional organisations, serving as president of the Association of Symbolic Logic from 1956 to 1958. In 1961, he served as president of the International Union of the History and the Philosophy of Science. He died in Madison, Wisconsin.
Kleene pronounced his last name /klay'nee/. /klee'nee/ and /kleen/ are extremely common mispronunciations.
This article (or an earlier version of it) contains material from FOLDOC, used with permission.