Compact Kaehler manifolds are subject to
numerous topological restrictions, mainly coming from Hodge theory applied
to the underlying Riemannian manifolds. We shall review them and
explain the crucial notion of (polarized) Hodge structure.
On the other hand, projective complex varieties are the principal source of construction of compact Kaehler manifolds, and it has been believed, following Kodaira's work on surfaces, that any compact Kaehler manifold could be obtained by deforming the complex structure of a complex projective manifold.
We show in fact that Hodge theory provides topological obstructions for a compact Kaehler manifold to admit a projective complex structure, and we construct compact Kaehler manifolds which do not have the homotopy type of projective complex manifolds.