  We describe connections between concepts arising in Poisson geometry and the
theory of Fukaya categories. The key concept is that of a symplectic groupoid,
which is an integration of a Poisson manifold. The Fukaya category of a
symplectic groupoid is monoidal, and it acts on the Fukaya categories of the
symplectic leaves of the Poisson structure. Conversely, we consider a wide
range of known monoidal structures on Fukaya categories and observe that they
all arise from symplectic groupoids. We also use the picture developed to
resolve a conundrum in Floer theory: why are some Lagrangian Floer cohomology
rings commutative?
