Bolyai, who was born in Koloszvár (now Cluj) in Romania, was the son of Farkas Bolyai, a distinguished mathematician who had an obsession with the status of Euclid's famous parallel postulate and devoted his life to trying to prove it. Despite his father's warnings that it would ruin his health, peace of mind, and happiness, Janos too started working on this axiom until, in about 1820, he came to the conclusion that it could not be proved. He went on to develop a consistent geometry in which the parallel postulate is not used, thus establishing the independence of this axiom from the others. In 1882 Bolyai published an account of his non-Euclidean geometry. Although his discovery had been anticipated by Nikolai Lobachevsky and Karl Gauss he was unaware of their work.

The discovery of the possibility of non-Euclidean geometries had a tremendous impact on both mathematics and philosophy. In mathematics it opened the way for a far more general and abstract approach to geometry than had previously been pursued, and in philosophy it settled once and for all the arguments about the supposed privileged status of Euclid's geometry. Bolyai also did valuable work in the theory of complex numbers.