  We study codimension one holomorphic foliations on complex projective spaces
and compact manifolds under the assumption that the foliation has a projective
transverse structure in the complement of some invariant codimension one
analytic subset. The basic motivation is the characterization of pull-backs of
Riccati foliations on projective spaces. Our techniques apply to give a
description of the generic models of codimension one foliations on compact
manifolds of dimension $\ge 3$.
