We shall trace the early history of the Exponential Theorem that states that, in the algebra of the formal power series in two non-commuting ideterminates x, y, the series naturally associated to log(e^xe^y) is a series of Lie polynomials in x,y. This theorem is a result of considerable importance in the theory of Lie groups. It is surprising perhaps that it has a complicated history going back over a century and involving many mathematicians.