Symplectic Laplacians are introduced by Tseng and Yau in 2012, which are related to a system of supersymmetric equations from physics. These Laplacians behave different from usual ones in Rimannian case and Complex case. They contain both 2nd and 4th order operators. In this talk, we will discuss these operators and their relations with cohomologies on compact symplectic manifolds with boundary. For this purpose, we will introduce new boundary conditions for differential forms on symplectic manifolds. Their properties and importance will be discussed.