The propagation characteristic of optical vortex beams in the free space is studied. By using the Huygens-Fresnel formula and ABCD optical matrix, the propagation expression of the Laguerre-Gaussian beams in the free space is derived. The properties of vortex beams in several different rotationally symmetric optical systems are analyzed. The result shows that the rotationally symmetric optical system just influences the diffraction properties or radius of curvature of the beam, but does not influence the orbital angular momentum of the vortex beams.