  We prove that a holomorphic projective connection on a complex projective
threefold is either flat, or it is a translation invariant holomorphic
projective connection on an abelian threefold. In the second case, a generic
translation invariant holomorphic affine connection on the abelian variety is
not projectively flat. We also prove that a simply connected compact complex
threefold with trivial canonical line bundle does not admit any holomorphic
projective connection.
