  We show that in every dimension greater than or equal to 4, there exist
compact Kaehler manifolds which do not have the homotopy type of projective
complex manifolds. Thus they a fortiori are not deformation equivalent to a
projective manifold, which solves negatively Kodaira's problem.
  We give both non simply connected (of dimension at least 4) and simply
connected (of dimension at least 6) such examples.
