Weak approximation and non-abelian fundamental groups

We introduce a new obstruction to weak approximation which is related to et ale non-abelian coverings of a proper and smooth algebraic variety X define d over a number field k. This enables us to give some counter examples to w eak approximation which are not accounted for by the Brauer-Manin obstructi on, for example bielliptic surfaces. (C) 2000 Editions scientifiques et med icales Elsevier SAS.