Based on the generalized Huygens-Fresnel diffraction integral, the basic and transformation properties of flattened Gaussian beams are studied in an unified way, where the flattened Gaussian beam is regarded as a whole beam. The analytical propagation equations of flattened Gaussian beams through a paraxial ABCD optical system with linear gain (absorption), unapertured and apertured ABCD systems are derived. The similar transformation of flattened Gaussian beam and correspondency between flattened Gaussian beams and super-Gaussian beams are analyzed. The results provide a useful simulation model for high-power lasers.