The Kontsevich formality theorem, established more than 20 years ago, implies that every Poisson manifold has a formal deformation quantization. The existence of non-formal deformations, on the other hand, remains largely an open problem. We study star products defined by semi-classical integral Fourier operators. Our main result states that a Poisson manifold which admits such a star product must be integrable by a symplectic groupoid. This is on-going joint work with Alejandro Cabrera (UFRJ).
