Abstract: In the paper,some important issues while analyzing general astigmatic Gaussian(GAG) beams are stressed on.Then typical numerical examples are given to illustrate the transformation properties of GAG beams,i.e. for GAG beams,the waist is not certainly located at the position corresponding to the infinite curvature radius of the wave-front and there may be several waists even though in free space,which are different from those of conventional stigmatic Gaussian and simple astigmatic Gaussian beams.