Helmholtz equation arises in many problems related to wave propagations, such as acoustic, electromagnetic wave scattering and in geophysical applications. Developing efficient and highly accurate numerical schemes to solve Helmholtz equation at large wave numbers is a very challenging scientific task and it has attracted a great deal of attention for a long time. The foremost difficulty in solving Helmholtz equation numerically is to eliminate or minimize the pollution effect which could lead to a serious problem as the wave number increases. Let k, h, and n denote the wave number, the grid size and the order of a finite difference or finite element approximations, it can be showed that the relative error is bounded by , where  or 1 for a finite difference or finite element method. It has been reported that it is impossible to eliminate the pollution effect that in two and more space dimensions. Recently, new finite difference schemes are developed for one-dimensional Helmholtz equation with constant wave numbers, and it has been verified that error estimate is bounded by  and the convergence is independent of the wave number k even when kh >1. In this talk, we extend the idea on constructing the pollution –free difference schemes to multi-dimensional Helmholtz equation in the polar and spherical coordinates. The superior performances of the new schemes are validated by comparing the numerical solutions with those obtained by the standard finite difference and the fourth-order compact schemes. The new scheme can also be applied to certain problems in a rectangular domain.