In this talk, I will introduce new cohomologies and elliptic operators on symplectic manifolds. Their construction follows from a simple decomposition of the exterior derivative into two first-order symplectic differential operators, which are analogous to the Dolbeault operators in complex geometry. These symplectic cohomologies encode new geometrical invariants especially for non-Kahler symplectic manifolds. This is joint work with S.-T. Yau.