  This paper is intended both an introduction to the algebraic geometry of
holomorphic Poisson brackets, and as a survey of results on the classification
of projective Poisson manifolds that have been obtained in the past twenty
years. It is based on the lecture series delivered by the author at the Poisson
2016 Summer School in Geneva. The paper begins with a detailed treatment of
Poisson surfaces, including adjunction, ruled surfaces and blowups, and leading
to a statement of the full birational classification. We then describe several
constructions of Poisson threefolds, outlining the classification in the
regular case, and the case of rank-one Fano threefolds (such as projective
space). Following a brief introduction to the notion of Poisson subspaces, we
discuss Bondal's conjecture on the dimensions of degeneracy loci on Poisson
Fano manifolds. We close with a discussion of log symplectic manifolds with
simple normal crossings degeneracy divisor, including a new proof of the
classification in the case of rank-one Fano manifolds.
