Years ago representation theory was a very specialized field, and very few non-specialists had much interest in it. This situation has changed profoundly in recent times. Due to the efforts of people like Gelfand, Harish-Chandra, Lang- lands, Witten, and others, it has come to occupy a central place in contemporary mathematics and theoretical physics. This talk takes a brief look at the myriad ways that representations of groups enters mathematics and physics. It turns out that the evolution of this subject is tied up with the evolution of the concept of space itself and the classifi cation of the groups of symmetries of space. This talk takes a brief look at the myriad ways that representations of groups enters mathematics and physics. It turns out that the evolution of this subject is tied up with the evolution of the concept of space itself and the classification of the groups of symmetries of space.