This paper provides a brief history of B-series and the associated Butcher
group and presents the new theory of word series and extended word series. B-series
(Hairer and Wanner 1976) are formal series of functions parameterized by rooted trees.
They greatly simplify the study of Runge-Kutta schemes and other numerical integrators.
We examine the problems that led to the introduction of B-series and survey a number of
more recent developments, including applications outside numerical mathematics. Word
series (series of functions parameterized by words from an alphabet) provide in some
cases a very convenient alternative to B-series. Associated with word series is a group G
of coe cients with a composition rule simpler than the corresponding rule in the Butcher
group. From a more mathematical point of view, integrators, like Runge-Kutta schemes,
that are a ne equivariant are represented by elements of the Butcher group, integrators
that are equivariant with respect to arbitrary changes of variables are represented by
elements of the word group G.
