Noncommutative field theories from a deforma - tion point of view

In this review we discuss the global geometry of noncommutative field theories from a deformation point of view: The space-times under consideration are deformations of classical space-time manifolds using star products. Then matter fields are encoded in deformation quantizations of vector bundles over the classical space-time. For gauge theories we establish a notion of deformation quantization of a principal fiber bundle and show how the deformation of associated vector bundles can be obtained. Mathematics Subject Classification (2000). Primary 53D55; Secondary 58B34, 81T75.