Abstract: Kähler manifolds are Hermitian manifolds with a compatible symplectic structure. They are exactly the Riemannian manifolds of dimension 2n with holonomy contained in U(n). An important subclass are the smooth complex projective varieties. Using Hodge theory, it turns out that on Kähler manifolds the Hodge cohomology groups allow a decomposition into the Dolbeault cohomology, which, together with Poincare and Serre duality, leads to relations between the Betti- and Hodge-numbers and consequently obstructions to the existence of a Kähler metric. We recall the definitions and basic properties of Dolbeault cohomology and Kähler manifolds and retrace the main steps in the proof of the above mentioned decomposition.