In this chapter, we present some rudiments of what is sometimes called multi-Hamiltonian geometry, which, roughly speaking, is the theory of the manifolds endowed with two or more Poisson structures subjected to suitable compatibility conditions. To explain the significance of this class of manifold in the theory of classical integrable systems, we recall that the theorem of Arnold-Liouville-Mineur does not provide any insight into how to find the first integrals of a given Hamiltonian systems, which are necessary to check its complete integrability. For this reason, it becomes important to look for additional structures that could provide a given Hamiltonian vector field with a sufficiently rich supply of first integrals in involution. As we will see in this chapter, the existence of multi-Hamiltonian representation of a dynamical system, induced by a family of compatible Poisson structures, implies the appearance of a set of first integral in involution, which, if further suitable conditions are verified, would imply the integrability of the dynamical system.