  We discuss the prominence of Hopf algebras in recent progress in Quantum
Field Theory. In particular, we will consider the Hopf algebra of
renormalization, whose antipode turned out to be the key to a conceptual
understanding of the subtraction procedure. We shall then describe several
occurences of this or closely related Hopf algebras in other mathematical
domains, such as foliations, Runge Kutta methods, iterated integrals and
multiple zeta values. We emphasize the unifying role which the Butcher group,
discovered in the study of numerical integration of ordinary differential
equations, plays in QFT.
