Abstract: On a symplectic manifold one can define a co-differential, a linear first order differential operator mapping differential forms of degree k to differential forms of degree k-1. In an attempt to do 'symplectic Hodge theory' Brylinksi asked if every deRham cohomology class on a compact symplectic manifold has a representative which is also co-closed. Though formally this looks very similar to the Riemannian analogue, the symplectic situation is essentially different since the 'symplectic Laplacian' vanishes and one has no elliptic theory at hand to attack the problem. Mathieu showed that a symplectic manifold has the Brylinksi property iff it satisfies the Hard Lefschetz Theorem. For general symplectic manifolds one can thus asked which cohomology classes admit co-closed representatives. In the talk we will discuss how to compute these 'symplectically harmonic' cohomology classes and derive some nice properties.