Abstract: On the basis of spherical vector wave functions(SVWF) in homogeneous gyrotropic anisotropic medium, the first, second spherical Bessel functions are given, satisfying the same differential equation and recursive formula, the electromagnetic fields in two concentric gyrotropic anisotropic spheres, and free space can be expressed in terms of SVWFs in anisotropic gyrotropic medium and free space. Applying the continue boundary conditions of electromagnetic fields in the interface of the gyrotropic anisotropic spherical shell, the coefficients equations of electromagnetic fields in terms of SVWF in two concentric gyrotropic anisotropic spheres and scattering fields in free space are derived. Numerical results for the very general gyrotropic dielectric media are obtained and those in a special case are compared between the present method and the homogeneous gyrotropic sphere where the parameters of two concentric gyrotropic anisotropic spheres are the same.