Some of the most basic invariants on manifolds come from cohomologies of differential forms.  In this talk, I will introduce a number of new cohomologies on symplectic manifolds.  Their construction follows simply from a symplectic decomposition of the exterior derivative operator into two linear differential operators, which are analogous to the Dolbeault operators in complex geometry.  Associated with the cohomologies are new elliptic operators which exhibit Hodge theoretical properties.  The cohomologies encode novel symplectic invariants that are especially interesting for non-Kahler symplectic manifolds.