Bolyai was born in Kolozsvar (today Cluj-Napoca in Romania), Transylvania, at that time part of the Austro-Hungarian Empire. When he was 13, he had mastered the calculus and other forms of analytical mechanics, his father Farkas Bolyai giving him instructions.
He studied at the Royal Engineering College in Vienna from 1818 to 1822. Between 1820 and 1823 he prepared a treatise on a complete system of non-Euclidean geometry. Bolyai's work was published in 1832 as an Appendix to an essay by his father.
Gauss, on reading the Appendix, wrote to a friend saying "I regard this young geometer Bolyai as a genius of the first order". In 1848 Bolyai discovered that Lobachevsky had published a similar piece of work in 1829.
In addition to his work in geometry, Bolyai developed a rigorous geometric concept of complex numbers as ordered pairs of real numbers. Although he never published more than the 24 pages of the Appendix, he left more than 20000 pages of manuscript of mathematical work when he died. These are now in the Bolyai-Teleki library in Targu-Mures, Romania.
He was an accomplished linguist speaking nine foreign languages including Chinese and Tibetan. In Cluj-Napoca a university is called after him, the Babes-Bolyai University.