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