So, if I understand correctly, theory says space is curved? Assuming my understanding is correct, have scientists ever explained how they can see so far across space withing hitting a horizon? (Or, if my premise is off-base, please hellp me better understand.)
Well, for one thing, space is very very big. Imagine living on a planet umpty-billion light-years in diameter. The horizon would be a long long ways away.
But even more relevant: A curved space takes a little understanding to wrap one’s mind around. Any object in space can be curved (think of a hula hoop, or a floppy disk left sitting in the sun too long). That doesn’t mean space itself is curved.
The notion that space itself is curved means everything in that space curves along with it. The usual analogy is this: Think of a sheet of soft plastic with geometrical shapes drawn on it. Now stretch and bend the plastic. Everything drawn on the sheet stretches and bends along with it. Curved space is understood to be something like that.
Not only do all objects in space bend and curve along with the space itself, but even light does. Imagine Vladimir Putin in Moscow trying to send a message to Queen Elizabeth in London by blinking a flashlight in a westward direction. But imagine further that, as the light zips along, it follows the curve of the earth’s surface instead of going straight off into outer space. If you have a beam of light zipping through curved interstellar space, it follows the curve of the space it is in like that. So in that sense, there are no horizons at all. You could see infinitely far.
To be sure, when you bring in “time” as the “fourth dimension”, then things get different. In the 4-dimensional space-time universe, there is a horizon, having to do with how far away you can see (in space-time) before you bump into the space-time horizon (commonly called the “event horizon”). But I think that whole discussion is outside the OP’s question, besides being well-discussed to death elsewhere and elsewhen.
You only see a horizon if you are above the curved surface. If you are inside it, it is like how light behaves in a fibre-optic cable - no matter how it twists the light goes “straight.”
At the moment we’ve measured the cosmological curvature of space and it’s actually, very flat, in fact it’s so flat it’s a slight problem for basic big bang theory.
On Earth the fact that our viewing distance is limited by the horizon is because the Earth obstructs our view. Space doesn’t obstruct the view of space. It’s very difficult to conceive of a horizon in space that it is even remotely analagous to the Earth’s horizon.
There are horizons in cosmological space, for example the particle horizon which is how far away we can see in theory and defines the size of the observable Universe. However this is not directly related to the curvature of space, though it is governed by the same factors.
What about the holographic principle? If that’s true, wouldn’t that answer the OP’s question inasmuch as the edge of the universe would probably be the “horizon?”
Well if space is curved, light travels in curved paths so you wouldn’t see a horizon.
Great word. Yours?
Senegoid may (or may not) have independently coined the word, but “elsewhen” is in relatively common usage among relativists.
And the large-scale curvature of the Universe is very, very close to flat, but there’s definite curvature on smaller scales. By way of analogy, consider a sheet of egg-crate foam: On a large scale, it’s a flat sheet, but on a small scale, you’ve got all sorts of wiggles and ripples.
What’s a floppy disk, grandpa?
It’s a DVD that’s been left in the sun on your dashboard too long.
Right, I didn’t invent that at all. Also, since the time dimension is sometimes seen as a stream that flows in one direction, the words “upwhen” and “downwhen” are also sometimes used.
“What’s a DVD?” said the fifth grader whose only experience of at-home movies is through streaming, whose only experience of purchased computer games and software is through downloads, whose only experience of music players is ipod-type devices, and whose only experience of portable data is thumb nail drives.
Thanks, all for broadening my horizons! 
If you started this thread just to make that joke, I’m going to hunt you down and make sausage from your dangly bits, then make you eat it.
Or salute you; I haven’t decided yet.
One thing to note though generally speaking it only makes sense to talk about spacetime curvature rather than spatial curvature. On cosmological scales though you can talk about purely spatial curvature (not that all the spacetime curvature can be expressed in terms of spatial curvature, hence cosmological expansion).
This is because in a homogeneous and isotropic space, spacetime curvature can’t instantaneously affect objects with different velocities in different ways.
As some may have noticed, I collect words. I must have ten from physicists alone in SD. What else do you guys have up your sleeve? 
Okay, let’s invent a few right here and now.
Any writer (especially journalists) know the famous questions: Who? What? When? Where? Why? and How? And we have “elsewhere” and “elsewhen”.
So let’s just add: elsewho (and elsewhom), elsewhat, elsewhy, and elsehow.
ETA: So, Leo Bloom, what are your ten nifty elsewords?
I’ve always liked “light cone.”
Physicists tend to use words like “chronological future” and “causal future” and “achronal” (all applied to sets of events in spacetime).
It is in the context of discussing light cones that the words “upwhen” and “downwhen” sometimes appear.
To be overly fastidious, the ‘light cone’ is usually used to refer to a subset of the tangent space of an event, rather than a subset of spacetime. In flat spacetime the difference between the tangent space of an event and the spacetime itself is more of technicality, but in curved spacetime you can’t ignore the difference. I never heard of ‘up when’, but I guess it would correspond to the ‘causal future’ when used in reference to a particular event.