Galilean invariance is one of the key requirements of many physical models adopted in theoretical and computational mechanics. Recent research developments in shock hydrodynamics computations [1] revealed the need for a detailed invariance analysis for SUPG operators. Lack of Galilean invariance can yield catastrophic instabilities in Lagrangian computations. The analysis is developed by means of an arbitrary Lagrangian-Eulerian (ALE) framework [2,3,4], in which stabilization operators for Lagrangian and Eulerian mesh computations are obtained as limits of the stabilization operator for the underlying ALE formulation. In the case of Eulerian meshes, it was discovered [2,3,4] that most of the SUPG operators designed for compressible flow computations to date are not consistent with Galilean invariance. In addition, Galilean invariant stabilized formulations can provide consistent advantages in complex engineering applications, due to the simple modifications needed for their implementation.