In this talk, we will introduce a systematic scheme for explicitly constructing Green’s function for an initial-boundary value problem of evolutionary partial differential equations. To illustrate our approach, we first solve some basic PDEs such as the heat equation, the wave equation, and the damped wave equation, in a half-space and a quarter plane with various boundary conditions. Moreover, we will introduce the forming of the surface wave for these equations with particular boundary conditions. Then we will construct the complete representations of Green’s functions for the convection-diffusion equation and the drifted wave equation in a half-space with various boundary conditions. Finally, we will provide some ideas for constructing Green’s function of hyperbolic-dissipative system such as the compressible Navier-Stokes equations.