The Milky Way galaxy is moving through space at a speed of approximately 552 kilometers per second (343 miles per second) relative to the cosmic microwave background radiation (CMB). This is the standard reference frame for measuring the motion of galaxies.

However, it's important to note that the Milky Way is also gravitationally interacting with other galaxies in the local group, like Andromeda. So, it's also moving towards Andromeda at a speed of around 110 kilometers per second (68 miles per second).

Overall, the Milky Way's motion is complex and constantly changing due to its interactions with other galaxies and the expanding universe.