by the Poisson bracket on . More precisely, a Poisson manifold consists of a smooth manifold (not necessarily of even dimension) together with an abstract bracket , still called Poisson bracket, which does not necessarily arise from a symplectic form , but satisfies the same algebraic properties. Poisson geometry is closely related to symplectic geometry: for instance every Poisson bracket determines a foliation of the manifold into symplectic submanifolds. However, the study of Poisson geometry requires techniques which are usually not employed in symplectic geometry, such as the theory of Lie groupoids and algebroids. Moreover, there are natural examples of structures which should be "morally" symplectic, but exhibit singularities, i.e. their "symplectic form" should be allowed to be degenerate. For example, the smooth quotient of a symplectic manifold by a group acting by symplectomorphisms is a Poisson manifold, which is general is not symplectic. This situation models the case of a physical system which is invariant under symmetries: the “reduced” phase space, obtained quotienting the original phase space by the symmetries, in general is no longer symplectic, but is Poisson. History Although the modern definition of Poisson manifold appeared only in the 70's-80's, its origin dates back to the nineteenth century. Alan Weinstein synthetised the early history of Poisson geometry as follows:"Poisson invented his brackets as a tool for classical dynamics. Jacobi realized the importance of these brackets and elucidated their algebraic properties, and Lie began the study of their geometry." Indeed, Siméon Denis Poisson introduced in 1809 what we now call Poisson bracket in order to obtain new integrals of motion, i.e. quantities which are preserved throughout the motion. More precisely, he proved that, if two functions and are integral of motions, then there is a third function, denoted by , which is an integral of motion as well. In the Hamiltonian formulation of mechanics, where the dynamics of a physical system is described by a given function (usually the energy of the system), an integral of motion is simply a function which Poisson-commutes with , i.e. such that . What will become known as Poisson's theorem can then be formulated asPoisson computations occupied many pages, and his results were rediscovered and simpliﬁed two decades later by Carl Gustav Jacob Jacobi. Jacobi was the ﬁrst to identify the general properties of the Poisson bracket as a binary operation. Moreover, he established the relation between the (Poisson) bracket of two