The best way too think of this is the balloon analogy. The dots on the balloon are, say, galaxies or galactic clusters. As the balloon inflates, they are moving away from each other on the surface.
Now, this is a three dimensional model of our four dimensional world, so we have had cut out one of the dimensions. People living on the surface of the balloon can only move in two spatial dimensions instead of three. They are constrained only to have [as we have on the surface of the earth roughly] only two degrees of freedom: the polar and azimuthal angles θ and φ of spherical coordinates, or, what is more familiar to laymen, latitude and longitude. The third dimension on this balloon, the distance from the center of the balloon, is time.
This is a very old analogy. There is a story about Einstein that goes with it.
When Einstein escaped from the Nazis, he worked at the center for advanced studies in Princeton, New Jersey, and he used to go for walks to think his deep thoughts. In the course of those walks he had a favorite little ice cream shop he used to go into, and the store owner had heard about the analogy of the balloon, and he asked Einstein essentially the question that you're asking: "What's outside of the sphere of our universe?"
Reportedly, Einstein took a couple of licks of the ice cream, thought for a moment, and then said, in his heavy German accent, "Yah... Vell... chust don't chew go out dere."
Now, think about what is inside of the balloon, and it will tell you what is outside of the balloon. And it will also tell you what is on the outside of the four dimensional balloon of our universe.
I'll post the answer tomorrow. [But better: you post the answer to me first.]
Well, about the only answer I can come up with is that the universe is expanding outward as a result of the "bang" and the space between the point where the bang occurred and where the matter now sits, is just empty space like the inside of the balloon.
As far as the expanding balloon we are on, there is nothing beyond it but dark, empty space.........