On the basis of Fresnel diffraction theory, the propagation properties of Laguerre-Gaussian vortex beam of any-order in free space were strictly deduced by using the Collins formula.The invariance of the expression form of Laguerre-Gaussian vortex beam of any-order in free space was proved.Only the case of free space propagation of Laguerre-Gaussian vortex beam was conoerned in the paper, nevertheless, the results can be extended to any paraxial optical system.Meanwhile, the phase and intensity distributions of Laguerre-Gaussian vortex beam of different order were numerically simulated.The simulation shows that after propagation, the phase and intensity distributions are both changed.These changes were further analyzed in virtue of the propagation expression.