  The purpose of this paper is to present a ``Cech-De Rham'' model for the
cohomology of leaf spaces. This model lends itself to the construction of
characteristic classes (in the cohomology of classifying spaces) by explicit
geometrical constructions which are immediate extensions of the standard
constructions for manifolds. In particular we rediscover (and explain) the
Thurston formula and the Bott formulas for cocycles on diffeomorphism groups.
We also use the Cech-De Rham model to explicitly describe the relation between
the cohomology of the classifying space, the basic cohomology, and the foliated
cohomology of foliations.
