Abstract: Holomorphic Poisson structures on projective space and other Fano manifolds play an important role in noncommutative algebra and generalized complex geometry, but relatively little is known about their classification. One of the difficulties is that the space of deformations of a given Poisson bracket is highly sensitive to the singularities of the bracket. In recent joint work with Travis Schedler, we introduced a natural new nondegeneracy condition on the singularities of Poisson brackets, called holonomicity. It allows the deformation space to be determined using topological tools such as perverse sheaves and intersection cohomology. I will give an introduction to the geometry of holonomic Poisson structures, and describe some applications to the classification of Poisson Fano manifolds.