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The multiloop amplitudes for open bosonic string in presence of a constantB-field are derived from first principles. The basic ingredients of theconstruction are the commutation relations for the string modes and the Reggeonvertex describing the interaction among three generic string states. Themodifications due to the presence of the B-field affect non--trivially only thezero modes. This makes it possible to write in a simple and elegant way thegeneral expression for multiloop string amplitudes in presence of a constantB-field. The field theory limit of these string amplitudes is also considered.We show that it reproduces exactly the Feynman diagrams of noncommutative fieldtheories. Issues of UV/IR are briefly discussed.Comment: based on talks and lectures presented at Summer Workshop on Branes  (KIAS, Seoul, June 2000); RTN-workshop "The Quantum Structure of Spacetime  and the Geometric Nature of Fundamental Interactions" (Berlin, Germany,  October 2000); Euroconference "Brane New World and Noncommutative Geometry"  (Villa Gualino, Torino, Italy, October 2000). v2, more discussions on the  relation between string coulplings, references adde

MULTILOOP NONCOMMUTATIVE OPEN STRING THEORY AND THEIR QFT LIMIT