Regiomontanus, as befitted a Renaissance humanist, changed his name, Johann Müller, for a Latin version of the name of his home town Königsberg (now Kaliningrad in Russia). His father was a miller. He was educated at Leipzig and Vienna where he was a pupil of Georg von Purbach. One of the ambitions of Purbach had been to produce a good text of Ptolemy's Almagest based on the original Greek rather than translations from Arabic at third or fourth hand. He had intended to go to Italy in quest of manuscripts of Ptolemy and other ancient scientists with the great Greek scholar Cardinal Bessarion. The death of Purbach in 1461 allowed Regiomontanus to take his place and spend six years in Italy searching for, translating, and editing manuscripts. After his return he settled in Nuremberg where his wealthy benefactor, Bernard Walther, built him an observatory and provided him with instruments. In 1475 he was called to Rome by the pope, Sixtus IV, to help in the reform of the calendar but died of the plague (or, possibly, poison) in 1476.

Regiomontanus was one of the key figures of 15th-century science. In the 1460s he wrote De triangulis (On Triangles), a work not printed until 1533 but that was, together with Tabulae directionum (1475; Tables of Direction), the main channel for the introduction of modern trigonometry into Europe. In the latter work he broke away from the ancient tradition of chords and instead gave tables of sines for every minute and tangents for every degree.