There is a natural holomorphic Poisson structure on any flag manifold $G/P$ with infinitely many symplectic leaves, but only finitely many $T$-orbits of symplectic leaves. I'll describe these in detail for the case of the Grassmannian, where the leaves are naturally indexed by juggling patterns (this work is joint with Thomas Lam and David Speyer). Then I'll talk about manifolds with atlases consisting of Kac-Moody Bruhat cells, which conjecturally includes wonderful compactifications of groups.