  B-series originated from the work of John Butcher in the 1960s as a tool to
analyze numerical integration of differential equations, in particular
Runge-Kutta methods. Connections to renormalization theory in perturbative
quantum field theory have been established in recent years. The algebraic
structure of classical Runge-Kutta methods is described by the Connes-Kreimer
Hopf algebra. Lie-Butcher theory is a generalization of B-series aimed at
studying Lie-group integrators for differential equations evolving on
manifolds. Lie-group integrators are based on general Lie group actions on a
manifold, and classical Runge-Kutta integrators appear in this setting as the
special case of R^n acting upon itself by translations. Lie--Butcher theory
combines classical B-series on R^n with Lie-series on manifolds. The underlying
Hopf algebra combines the Connes-Kreimer Hopf algebra with the shuffle Hopf
algebra of free Lie algebras. We give an introduction to Hopf algebraic
structures and their relationship to structures appearing in numerical
analysis, aimed at a general mathematical audience. In particular we explore
the close connection between Lie series, time-dependent Lie series and
Lie--Butcher series for diffeomorphisms on manifolds. The role of the Euler and
Dynkin idempotents in numerical analysis is discussed. A non-commutative
version of a Faa di Bruno bialgebra is introduced, and the relation to
non-commutative Bell polynomials is explored.
