  We construct real polarizable Hodge structures on the reduced leafwise
cohomology of K\"ahler-Riemann foliations by complex manifolds. As in the
classical case one obtains a hard Lefschetz theorem for this cohomology.
Serre's K\"ahlerian analogue of the Weil conjectures carries over as well.
Generalizing a construction of Looijenga and Lunts one obtains possibly
infinite dimensional Lie algebras attached to K\"ahler-Riemann foliations.
Finally using $(\mathfrak{g},K)$-cohomology we discuss a class of examples
obtained by dividing a product of symmetric spaces by a cocompact lattice and
considering the foliations coming from the factors.
