Abstract: In this paper， we are devoted to a review of the Green functions and the Green

　　function method for solving the initial and boundary value problems of various ordinary

　　differential equations. For the initial value prob- lem of linear ordinary differential equation

　　of first order， the formula of Green function and the solution represented

　　by Green function are given. For the boundary value problem and initial value problem of linear

　　ordinary differential equation of second order， the calculation of Green function and the

　　solution represented by Green function are shown， respectively. The calculation of Green

　　function and the solution representation by Green function for Sturm-

　　Liouville problem satisfying general initial value conditions are established. For the

　　un-mixed and mixed boundary value problems of linear ordinary differential equations of second

　　order， we also consider how to find Green function and the solution represented by Green

　　function， respectively. Finally， For the boundary value problem of higher or- der linear ordinary

　　differential equations， the Green function and the solution expressed by Green function

　　are ob-

　　tained.

Key words: Green function, solution represented by convolution, linear superposition principle, 　　boundary val- ue problem, initial value problem