In 1897 G. Frobenius described for the first time the now familiar notion of the matrix representations of a group, and explained their relation to group characters. The new theory gave remarkable and deep results in finite group theory. But it soon became clear that the representations of classical groups were involved in many problems of harmonic analysis, geometry and quantum physics. Group representations continue to be involved in many, strikingly varied, advances in mathematics and theoretical physics in the present day.