After reviewing some basic properties of holomorphic Poisson geometry, we will present a decomposition result in the Kähler case: if a compact Kähler Poisson manifold has a compact symplectic leaf with finite fundamental group, then after passing to a finite étale cover, it splits as the product of the universal cover of the leaf and some other Poisson manifold. This can be viewed as a global analogue to a theorem due to Alan Weinstein describing local Poisson structures. Joint work with Stéphane Druel, Jorge Pereira and Brent Pym.