Many basic properties of manifolds can be obtained from studying their differential forms. In this talk, I shall describe the particular characteristics that differential forms have on symplectic manifolds. In the presence of a symplectic structure, I will show that the exterior derivative has a simple decomposition into two first-order differential operators analogous to that in complex geometry. Using this property, I will construct new symplectic cohomologies and elliptic operators that encode interesting geometrical invariants especially for non-Kahler symplectic manifolds.