  In this expository article we review recent advances in our understanding of
the combinatorial and algebraic structure of perturbation theory in terms of
Feynman graphs, and Dyson-Schwinger equations. Starting from Lie and Hopf
algebras of Feynman graphs, perturbative renormalization is rephrased
algebraically. The Hochschild cohomology of these Hopf algebras leads the way
to Slavnov-Taylor identities and Dyson-Schwinger equations. We discuss recent
progress in solving simple Dyson-Schwinger equations in the high energy sector
using the algebraic machinery. Finally there is a short account on a relation
to algebraic geometry and number theory: understanding Feynman integrals as
periods of mixed (Tate) motives.
