(Codimension-one) foliations by complex leaves are, roughly speaking, smooth partitions of a smooth manifold into complex manifolds of codimension-one (think of $\Bbb C^n\times \Bbb R$ foliated by the level sets of the projection map onto $\Bbb R$). They are mixed objects, tangentially holomorphic and transversally smooth.

They appear in different situations. Examples include differentiable families of deformations of compact complex manifolds, Levi-flat hypersurfaces in complex projective space, every oriented 2-dimensional smooth foliation. But there exist also more exotic ones as a foliation of the 5-sphere by complex surfaces.

They can be seen as generalized deformations of complex manifolds, or as generalized complex structures on odd-dimensional smooth manifolds.

In this talk, I will explain the notion of foliations by complex leaves and describe (some of) the previous examples in detail. Then, taking the viewpoint of generalized deformations, I will compare classical deformations and foliated deformations in the particular example of Hopf manifolds.

This is a joint work with M. Nicolau and A. Verjovsky.