Character groups of Hopf algebras have provided a convenient setting to formulate renormalisation procedures. Already in the earliest works by Connes and Kreimer on these topics, certain differential equations were formulated and solved on these groups. However, it was not until much later, that the global analytic structure of these groups was described. In this talk we will review the (infinite-dimensional) Lie group structure of character groups of Hopf algebras and certain subgroups. We will then report on their relation to renormalisation and recent developments and applications of these groups.