The propagation properties of laser beams are of much significance in applications of laser technology. The analytical propagation equation for beams is one of the focuses in laser optics. Starting from the Collins formula, the properties of the off-axis Gaussian beam (OAGB) propagating through a paraxial ABCD optical system with an aperture are studied in detail. Moreover, the analytical equations for the off-axis Gaussian beam propagating through a paraxial ABCD optical system with hard-edged aperture, Gaussian aperture or cosine aperture are derived. The relationship among hard-edged aperture, Gaussian aperture and cosine aperture is discussed. The validity and advantage of results are confirmed and illustrated by numerical calculations. The difficulty of finding the analytical propagation formula of laser beams propagating through a paraxial ABCD optical system with multi-aperture is analyzed. It is shown that the difficulty is due to the existence of an error function in the analytical propagation formula for single-apertured laser beams.