We introduce two generalizations of the Bethe ansatz that allows to treat the case of open spin chains with non diagonal boundary matrices. The first one is a generalization of the original coordinate Bethe ansatz, and the second one a generalization of the Matrix anstaz, used in statistical physics. The two ansaetzen are complementary in the sense that they provide different eigenvalues of the same Hamiltonian. Altogether they are conjectured to give the full spectrum. We illustrate it on two cases: the XXX and XXZ chains. Relations with algebraic Bethe ansatz will be also discussed.