Simple as that really.
In relation to what?
By Mach’s Principle, as a whole, no. At least in that if the distant stars were rotating relative to us, we would call it us rotating relative to them instead (and feel things like centrifugal force that would convince us that we were right).
I often hear the universe is “expanding”. I never understood what it’s expanding into, but if it’s expanding then why not revolving too?
My old cosmology prof brought this up. He proposed the following thought-experiment: suppose the entire cosmos is empty – no stars, no galaxies, no dust clouds, no microwave background – except for the earth, just sitting there, totally anomalous. Is it rotating? He said to check if it has an equatorial bulge. If it does, then, yes, it is rotating. In relation to what? “Space/time.”
Apparently, there are at least some mathematical models of “space/time” that imply a metric. Dimensions. Directions. Not absolute, necessarily, but intrinsic.
The same equations could suggest that this lonely earth is not rotating, but that the (empty) cosmos is rotating around it, and it’s “frame dragging” that causes the equatorial bulge.
He told us that there are other sets of equations that don’t give dimensions or extension or directions in empty space, and that the question is meaningless.
This has been talked about a lot on the “Bad Astronomy” discussion boards.
The usual short explanation is that the universe is not expanding into anything. It is all there is. It just is getting bigger. The analogy is to the surface of a balloon. This surface is two-dimensional: there is no inside, there is no outside, the surface is all there is and all there can be. Mathematically this is completely valid. If the surface of a balloon expands, it gets bigger but it doesn’t expand into anything. Any spot on the surface gets farther away from every other spot, but no spot is special and none is in the center.
Same thing for the universe, only in three dimensions. It gets bigger but no spot inside it is special and none is in the center. But if the universe revolved, that would imply there is a center (or a central axis) it is revolving around. Einstein showed that no point can be special in that way, though.
Obviously, as Trinopus points out, you can get a lot more technical than this. But it’s a way into thinking about it.
So there’s no “edge” to the universe, even in theory? If you went straight in one direction would you end up back at the same place, like on the balloon?
There’s no edge, even in theory. However, the current consensus is that the universe is flat rather than positively curved, so you’ll never end up in the same place. Doesn’t really matter: space can expand faster than light and has done so in the past, and you can’t travel even as fast as light. You’ll never catch up even in theory.
I don’t know if it represents “reality”, and I sure as Hell don’t understand its derivation, but doesn’t the Universe according to (logician) Kurt Gödel, rotate?
The “Gödel Universe” is allowed, by Einstein’s theory of general relativity, as a possible way that the Universe could behave. But our observations of the Universe tell us that the Universe does not, in fact, behave in this way. Too bad for the Universe, I suppose.
Is the universe rotating? Maybe
http://news.discovery.com/space/do-we-live-in-a-spinning-universe-110708.html
Rotation isn’t relative. If it were, we would be able to define a valid reference frame in which geostationary satellites work by magic, or one in which the material in a centrifuge has no reason to separate.
Note also that the Gödel Cosmology has some really weird properties, such as containing closed timelike loops (in other words, time travel would be possible).
This is only tangentially relevant, but I’m glad you asked.
Doesn’t rotation require an axis? Wouldn’t rotation be relative to the axis? I suppose the axis rotates as well, but there is a clear center to any rotation that I am aware of.
How can it be “flat” but with no edge, if it’s not infinite? (I’m assuming it can’t be infinite if it’s expanding..)
Nothing about physics makes intuitive sense. The math says that it’s flat. The math says there is no middle and no edge. If you want meaning, buy a dictionary. 
Funnily enough, the math that says it’s flat is not inconsistent with it being cylindrical.
It seems every time I read a thread on cosmology there’s a different answer.
I’ve just got used to all the knowledgeable people saying that the universe was infinite in extent right from the start, and is still infinite in extent, and is becoming less dense rather than bigger.
And that I must have misunderstood all the popular science books (written by real scientists) that said that the universe started out with all matter contained in an extremely small volume at the time of the Big Bang and has been expanding ever since.
And now you’re back saying that the universe is getting bigger. I don’t know what to think.