We introduce the notion of Haantjes algebra: It consists of an assignment of a familyof operator fields on a differentiable manifold, each of them with vanishing Haantjestorsion. They are also required to satisfy suitable compatibility conditions. Haantjesalgebras naturally generalize several known interesting geometric structures, arising inRiemannian geometry and in the theory of integrable systems. At the same time, aswe will show, they play a crucial role in the theory of diagonalization of operatorson differentiable manifolds. Assuming that the operators of a Haantjes algebra aresemisimple and commute, we shall prove that there exists a set of local coordinateswhere all operators can be diagonalized simultaneously. Moreover, in the general, non-semisimple case, they acquire simultaneously, in a suitable local chart, a block-diagonalform
