Perturbative quantum field theory yields asymptotic series of divergent integrals. The purpose of renormalization is to assign to these precise numerical values (finite parts) in a coherent manner. Physicists accomplished this quite successfully, in particular in quantum electrodynamical theory, but the “coherent manner” of choosing finite parts remains complicated. A. Connes and D. Kreimer propose a mathematically limpid and universal method to accomplish this : the Feynman diagrams which label the integrals are organized into a graded Hopf algebra, and the renormalized series stems from a Wiener-Hopf factorization of the asymptotic series, reinterpreted as a function (loop) $\gamma (\mu ,\varepsilon )$ with values in the associated formal group (depending on a unit mass $\mu $ linked to the grading, and a parameter $\varepsilon $ (dimension) used for an analytic continuation). In fact all these constructions come from a unique canonical operation on the Hopf algebra corresponding to the group of formal diffeomorphisms tangent to $\mathrm {Id}$.