Abstract: Propagation properties of decentered Hermite-Gaussian beams through misaligned first-order optical systems are studied by using the generalized Huygens-Fresnel diffraction integral, augmented matrix and Wigner distribution function methods. It is shown that, as a decentered Hermite-Gaussian beam passes through the misaligned first-order optical system, the closed property is preserved, the second-order moments matrix varies with the usual law, the beam propagation factor remains unchanged, the first order moments matrix evolves, as if it were a ray of geometrical optics.