  We describe the possible noncommutative deformations of complex projective
three-space by exhibiting the Calabi--Yau algebras that serve as their
homogeneous coordinate rings. We prove that the space parametrizing such
deformations has exactly six irreducible components, and we give explicit
presentations for the generic members of each family in terms of generators and
relations. The proof uses deformation quantization to reduce the problem to a
similar classification of unimodular quadratic Poisson structures in four
dimensions, which we extract from Cerveau and Lins Neto's classification of
degree-two foliations on projective space. Corresponding to the ``exceptional''
component in their classification is a quantization of the third symmetric
power of the projective line that supports bimodule quantizations of the
classical Schwarzenberger bundles.
