CHAPTER 3 INTRODUCES the spherical inclusion and inhomogeneity problems and pro-vides an explicit close form solution for the elastic fields. Eshelby’s original work focused on a general problem for ellipsoidal inclusion and inhomogeneity, which includes the solution in Chapter 3 as a special case. However, the general solution is given in terms of elliptic integrals, which is not straightforward for direct use. Therefore, this chapter will emphasize on the general methods toward the solution so that the readers can start with them for specific research problems and needs. First, the elastic Green’s function for general anisotropic elastic materials is derived in the Fourier integral form. For isotropic materials, it recovers the previous solution. Using it, the elastic solutions for ellipsoidal inclusion problems with a polynomial eigenstrain and body force are provided. This method has been extended to other sources in potential flow problems. The solution is used in solving the inhomogeneity problems for multiple ellipsoidal particles embedded in the infinite domain, so particle interactions can be considered. The interface continuity are discussed.