symplectic groupoid. **Step 1** is known as the **integrability problem** and was recently completely settled. Coste, Dazord and Weinstein in [@CDW1987] and independently Karasev in [@karasev1987] showed the existence of a local symplectic groupoid over any Poisson manifold, “local” meaning that the multiplication is defined only on a neighborhood of the unit space. Cattaneo and Felder in [@feldcat2000] gave an explicit construction of a topological groupoid canonically associated to any Poisson manifold, which is a global symplectic groupoid whenever the Poisson structure is integrable. Crainic and Fernandes in [@CF2001] derived an if and only if criterium which tells one when the previous construction yields a manifold. **Step 2** however was only partially achieved (see [@WX1991]).

If we compare this program with deformation quantization (see [@FFLS1997] and [@kontsevich1997]), we see that starting with an integrable Poisson manifold $P$ whose symplectic groupoid is $G$ we should have the following relation between objects involved in these programs:

                                  Deformation quantization           Quantization by symplectic groupoids
  ---------------------- ------------------------------------------ --------------------------------------
   Semi-classical level                      ?                                  $(G,G^{(m)})$
      Quantum level       $(C^{\infty}(P)[[\epsilon]],*_\epsilon)$           $(Q(G),Q(G^{(m)}))$

We can regard the symplectic groupoid over a Poisson manifold as a (semi-)classical version of the quantum algebra. In this picture $G^{(m)}$ should then correspond to a semi-classical version 