Classical mechanics can be formulated in general spaces in terms of a Poisson bracket of functions {f,g}. The abstract properties of such a bracket lead to the study of Poisson manifolds and their symmetries. A special case is where the bracket is non-degenerate and then it comes from a closed non-degenerate 2-form w. A pair (M,w) consisting of a manifold and such a 2-form is called a symplectic manifold.The geometry of symplectic manifolds is quite different from Riemannian geometry but shares some features with complex manifolds, especially Kaehler manifolds (which are a special case since the Kaehler form is closed and non-degenerate).

Symplectic manfiolds can be studied from many points of view. Their topology, differential geometry, symmetries have all been the object of much study in recent times.