Algebra, geometry, and topology of the substitution group of formal power series

A systematic description is given of properties of the group J(k) of formal power series in one variable with coefficients in a commutative unitary ring k. This topological group has been studied intensively over the past 20 years, and a number of interesting results on its structure have been obtained. Here it is indicated how the group J(k) arises in several different areas of mathematics, such as complex cobordism or symplectic topology. Also considered is how the general structure of the group of complex formal power series is connected with classical problems of local uniformisation and the embedding of the germ of a holomorphic map in a flow.