In this expository article we review recent advances in our understanding ofthe combinatorial and algebraic structure of perturbation theory in terms ofFeynman graphs, and Dyson-Schwinger equations. Starting from Lie and Hopfalgebras of Feynman graphs, perturbative renormalization is rephrasedalgebraically. The Hochschild cohomology of these Hopf algebras leads the wayto Slavnov-Taylor identities and Dyson-Schwinger equations. We discuss recentprogress in solving simple Dyson-Schwinger equations in the high energy sectorusing the algebraic machinery. Finally there is a short account on a relationto algebraic geometry and number theory: understanding Feynman integrals asperiods of mixed (Tate) motives.
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