of with coefficients in the trivial representation; the modular class of a Poisson manifold coincides with the modular class of the associated Lie algebroid . It is of crucial importance to notice that the Lie algebroid is not always integrable to a Lie groupoid. Symplectic groupoids A is a Lie groupoid together with a symplectic form which is also multiplicative, i.e. it satisfies the following algebraic compatibility with the groupoid multiplication: . Equivalently, the graph of is asked to be a Lagrangian submanifold of . Among the several consequences, the dimension of is automatically twice the dimension of . The notion of symplectic groupoid was introduced at the end of the 80's independently by several authors. A fundamental theorem states that the base space of any symplectic groupoid admits a unique Poisson structure such that the source map and the target map are, respectively, a Poisson map and an anti-Poisson map. Moreover, the Lie algebroid is isomorphic to the cotangent algebroid associated to the Poisson manifold . Conversely, if the cotangent bundle of a Poisson manifold is integrable to some Lie groupoid , then is automatically a symplectic groupoid. Accordingly, the integrability problem for a Poisson manifold consists in finding a (symplectic) Lie groupoid which integrates its cotangent algebroid; when this happens, the Poisson structure is called integrable. While any Poisson manifold admits a local integration (i.e. a symplectic groupoid where the multiplication is defined only locally), there are general topological obstructions to its integrability, coming from the integrability theory for Lie algebroids. Using such obstructions, one can show that a Poisson manifold is integrable if and only if it admits a complete symplectic realisation. The candidate for the symplectic groupoid integrating a given Poisson manifold is called Poisson homotopy groupoid and is simply the Weinstein groupoid of the cotangent algebroid , consisting of the quotient of the Banach space of a special class of paths in by a suitable equivalent relation. Equivalently, can be described as an infinite-dimensional symplectic quotient. Examples of integrations The trivial Poisson structure is always integrable, the symplectic groupoid being the bundle of abelian (additive) groups with the canonical symplectic form. A non-degenerate Poisson structure on is always integrable, the symplectic groupoid being the pair groupoid together with the symplectic form (for ). A Lie-Poisson structure on is always integrable, the symplectic groupoid being the (coadjoint) action groupoid , for the simply connected integration of