The son of a lawyer, Ulam was educated at the polytechnic in his native city of Lwow, Poland, where he completed his doctorate in 1933. Ulam quickly developed a reputation as an original mathematician and in 1936 he was invited to visit the Institute for Advanced Study, Princeton, for a year. He decided to remain in the United States and spent the period from 1937 to 1940 at Harvard before being appointed professor of mathematics at the University of Wisconsin. He became a naturalized American citizen in 1943.

At the same time Ulam was invited to work on the development of the atom bomb at Los Alamos, New Mexico, and he remained associated with Los Alamos until 1967. Here he worked on the theory of nuclear reactions. When neutrons are released in a reactor some scatter, some are absorbed, others escape or collide, etc. The actual process is too complex for analytical calculation. If, however, the fate of a practical number of neutrons is followed, and if at each branch one outcome with a suitable probability is selected, it is possible to derive a reasonably accurate mathematical model of the process. Ulam's technique, which is known as the ‘Monte Carlo method’ (after the casino), is a widely used numerical method in many different fields.

After the war Ulam worked with Teller on the development of the hydrogen bomb. He served as research adviser to the director of Los Alamos from 1957 to 1967, when he was appointed professor of mathematics at the University of Colorado, Boulder, a post he held until his retirement in 1977. He left an account of his life in his Adventures of a Mathematician (New York, 1976).