Moons orbit planets, planets orbit stars, and stars orbit the center of a galaxy.
So, my question is what does a galaxy orbit? The center of universe? (I know that the universe has no center)
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The term "orbit" means that an object moves around a point in space on a certain path. Generally, this center point is an object - like your examples - or, in a binary system, a point in space called a barycenter, around which both bodies orbit. The barycenter is located at the system's center of mass. Commonly cited examples of orbiting objects are the ones you gave above - moons, planets, and stars. But all of these objects orbit a center point.
If the universe has no center point, what can galaxies orbit? Each other? In some cases, this is true. The Milky Way is orbited by some smaller "satellite" dwarf galaxies, such as the Large and Small Magellanic Clouds. Andromeda, a nearby spiral galaxy, also has these satellites. But do the Milky Way and Andromeda orbit anything? No.
The Milky Way and Andromeda are part of the Local Group, our local galaxy cluster. They are gravitationally bound to each other. There is a center of mass in the Local Group, just as there is in a stellar system. But in the Local Group, the Milky Way and Andromeda are actually coming towards each other. A collision, and eventual merger, between the two, is expected to happen billions of years in the future.
On a larger scale, the Local Group is part of the Virgo Supercluster. It contains many other galaxy clusters. It, too, has a center of mass. But all the galaxy clusters are not "orbiting" this center of mass. They are merely bound together, thanks to gravity.
On an even larger scale, "filaments" are created out of long lines of galaxy superclusters. In fact, at even larger scales, the universe is thought to be homogenous - essentially the same in all directions. There is no single point in space that is the center, like you said, or that is different rom all the others.
Of course, the universe is thought to be expanding. In that case, not only are galaxy clusters and superclusters not closely orbiting each other, but the space between them is causing them to move further apart!
Also, you might find good explanations and/or related questions on the Astronomy Stack Exchange.
Generalizing the term "orbit" to mean some larger object / collection of objects to which the object in question is gravitationally bound, I'd say that the Milky Way "orbits" the Local Group, which in turn "orbits" the Virgo Supercluster.
Beyond that, the expansion of the universe starts to dominate over gravitation. There are larger structures than superstructures, but it's a big stretch to say that the members of those very, very large structures are orbiting one another.
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