If the universe is expanding, what is it expanding into ?

My closet.

More seriously, it isn't expanding into anything. The universe contains all of space-time. If it's expanding, this is a fact about the stuff that makes up the universe itself, and not about its relation to some other place/thing/void.

So what does it mean to say that the universe is expanding? At least this: that distances between things (on average) are getting larger. One common analogy: think about the surface of a balloon that's being blown up. Imagine the ballon as covered with little dots. Then as the balloon expands, each dot will get further away from each other dot.

The obvious objection is that the balloon is expanding into the space around it -- there's more than just the surface of the balloon. That's true. But that's why it's an analogy rather than a perfect fit. Although it puzzles the imagination, the math of space-time doesn't call for positing a container space for the expanding universe. The geometry is internal to the universe itself.

See also Question 619.

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