Abstract : Finding annihilators of the ideal class group of an abelian extension of Q is a classical subject which goes back to work of Kummer and Stickelberger. The Brumer-Stark conjecture deals with abelian extensions of number fields and predicts that a group ring element, called the Brumer-Stickelberger element, annihilates the ideal class group of the extension under consideration. Moreover it specifies that the generators thus obtained have special properties. The aim of this work is to generalize this conjecture to non-abelian Galois extensions. We first focus on the study of a non-abelian analogue of the Brumer element, necessary to establish a non-abelian generalization of the conjecture. The second part is devoted to the statement of our non-abelian conjecture, and the properties it satisfies. We are particularly interested in extension change properties. We then study the specific case of extensions whose Galois group has an abelian normal subgroup H of prime index. If the Brumer-Stark conjecture associated to certain abelian subextensions holds, we prove two results according to the parity of the cardinal of H : in the odd case, we get the non-abelian Brumer-Stark conjecture, and in the even case, we establish an abelianity result implying under additional hypotheses the proof of the non-abelian conjecture. Thanks to PARI-GP, we finally do some numerical verifications of the nonabelian conjecture, proving its validity in the tested examples.