Szilard, the son of an architect, studied engineering in his native city of Budapest before moving to the University of Berlin where he began the study of physics and obtained his doctorate in 1922. He remained there until 1933 when, after spending a few years in England working at the Clarendon Laboratory, in Oxford, and at St. Bartholomew's Hospital, London, he emigrated to America in 1938. After the war Szilard moved into biology and in 1946 was appointed to the chair of biophysics at the University of Chicago, where he remained until his death. He became a naturalized American in 1943.

Szilard was one of the first men in the world to see the significance of nuclear fission and the first to bring it to the attention of Roosevelt. In 1934, after hearing of the dismissal of the possibility of atomic energy by Ernest Rutherford, he worked out that an element that is split by neutrons and that would emit two neutrons when it absorbed one neutron could, if assembled in sufficiently large mass, sustain a nuclear chain reaction. Szilard applied for a patent, which he assigned to the British Admiralty to preserve secrecy.

When in 1938–39 he heard of the work of Otto Hahn and Lise Meitner on the fission of uranium he was well prepared. After quickly confirming that the necessary neutrons would be present Szilard, fearing the consequences that would ensue from Hitler's possession of such a weapon, decided that the only sound policy was for America to develop such a weapon first. To this end he approached Albert Einstein, with whom he had worked earlier and who commanded sufficient authority to be heard by all, and invited him to write a letter to the President of the United States. This initiated the program that was to culminate in the dropping of the atomic bomb on Hiroshima six years later. During the war Szilard worked on the development of the bomb and, in particular, worked with Enrico Fermi on the development of the uranium–graphite pile.

If Szilard was one of the first to see the possibility and necessity to develop the bomb he was also one of the earliest to question the wisdom and justice of actually using it against the Japanese. He was the dominant spirit behind the report submitted by James Franck to the Secretary of War in 1945 forecasting the nuclear stalemate that would follow a failure to ban the bomb.

Although his early reputation was based on his work in physics he moved, after the war, into molecular biology. Szilard took his new subject seriously, attending classes at the Cold Spring Harbor laboratories in 1946. He was soon to develop a high degree of competence, designing an important new instrument, the chemostat, formulating new theories on the aging process, and stimulating Jacques Monod in his work on the operon and the repressor.