  A tensorial approach to the theory of classical Hamiltonian integrable
systems is proposed, based on the geometry of Haantjes tensors. We introduce
the class of symplectic-Haantjes manifolds (or $\omega \mathscr{H}$ manifolds),
as a natural setting where the notion of integrability can be formulated. We
prove that the existence of suitable Haantjes algebras of (1,1) tensor fields
with vanishing Haantjes torsion is a necessary and sufficient condition for a
Hamiltonian system to be integrable in the Liouville-Arnold sense. We also show
that new integrable models arise from the Haantjes geometry. Finally, we
present an application of our approach to the study of the Post-Winternitz
system and of a stationary flow of the KdV hierarchy.
