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Nowadays, noncommutative geometry is a growing domain of Mathematics, which can also appear as a promising framework for modern Physics. Quantum field theories on "noncommutative spaces" are indeed thoroughly investigated and possess new interesting features. This book provides a pedagogical introduction to basic concepts of noncommutative geometry and deformation quantization, renormalization of quantum field theory: wilsonian and BPHZ approaches for scalar theory as well as algebraic approach for gauge theory. It allows the reader to understand several general problematics of noncommutative quantum field theory. As based on the PhD thesis of the author, this book gives an overview of the renormalization question for quantum field theories on the Euclidean Moyal space, and focuses particularly on the recent Grosse-Wulkenhaar model , its associated gauge theory, and their mathematical interpretation. This work is aimed at both postgraduate students and researchers who want to comprehend this lively research field at the frontier of Physics and Mathematics.

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Studies at the Ecole Polytechnique (France). PhD in Physics at the University of Paris XI (France) and in Mathematics at the University of Münster (Germany). Postdoctorant at the Catholic University of Louvain (Belgium).

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