Surfaces with constant anisotropic mean curvature (CAMC surfaces) are  surfaces which are in equilibrium for a constant coefficient parametric  elliptic functional with a volume constraint. In this talk, we explain  important properties of the Gauss map and the anisotropic Gauss map of CAMC  surfaces. Especially, we discuss a certain minimizing property and the exceptional set of the Gauss map for such surfaces. Also we give a remark about the harmonicity of the anisotropic Gauss map and some resent results.