  Exactly 170 years ago, the construction of the real quaternion algebra by
William Hamilton was announced in the Proceedings of the Royal Irish Academy.
It became the first example of non-commutative division rings and a major
turning point of algebra. To this day, the multiplicative group structure of
quaternion algebras have not completely been understood. This article is a long
survey of the recent developments on the multiplicative group structure of
division rings.
