How large is the universe? We know that the age of the universe (or, at least the time since the Big Bang) is roughly 13.75 billion years. I have heard that the size of the universe is much larger than what we can see, in other words, much larger than the observable universe. Is this true? Why would this not conflict with what is given by Special Relativity (namely, that faster-than-light travel is prohibited)? 
If it is true, then what is the estimated size of the universe, and how can we know?
 A: An snippet from the NASA page titled "How Big is Our Universe?":

No one knows if the universe is
  infinitely large, or even if ours is
  the only universe that exists. And
  other parts of the universe, very far
  away, might be quite different from
  the universe closer to home. Future
  NASA missions will continue to search
  for clues to the ultimate size and
  scale of our cosmic home.

A: This question has attracted a large number of incorrect answers.

We know that the age of the universe (or, at least the time since the Big Bang) is roughly 13.75 billion years. I have heard that the size of the universe is much larger than what we can see, in other words, much larger than the observable universe. Is this true? Why would this not conflict with what is given by Special Relativity (namely, that faster-than-light travel is prohibited)?

This part of the question involves two common misconceptions: (1) that the radius of the observable universe in light-years equals its age in years; and (2) that the Big Bang was an explosion that occurred at a point in empty space. Re misconception #1, see this answer. Re misconception #2, since the Big Bang was not an explosion that occurred at a point, there was no need for the exploding matter to travel outward at a speed higher than c in order to reach a certain distance away from us.

If it is true, then what is the estimated size of the universe, and how can we know?

First off, this leads to an issue in the philosophy of science. Although, by definition, we can't directly verify anything about the unobservable part of the universe, we can still infer things about it. Science basically always deals with extrapolation. We do a finite number of observations and experiments, and based on those, we infer general rules, which can be used to make predictions about things we haven't -- and possibly can't -- directly observe. We don't know that the sun will rise tomorrow, but we infer it based on a pattern of past observations. Similarly, we don't know that as time goes on we will continue to receive new information through our telescopes about parts of the universe as light from those regions reaches us. However, it's natural to infer that this process will continue to happen, based on the pattern of past observations. On similar grounds, we expect that since the presently observable universe is highly homogeneous and isotropic, the same will be true for the more distant parts that are not yet observable.
Based on these considerations, we construct cosmological models that are homogeneous and isotropic. These models fall into two categories, open and closed. Open models have negative spatial curvature and are spatially infinite. Closed models have positive spatial curvature and are spatially finite; they wrap around on themselves like a sphere (or possibly some other, more complex topology).
Finite models stay finite as they evolve over time, and infinite ones stay infinite. Therefore if the universe is infinite, there is no paradoxical need for matter to have traveled an infinite distance in a finite time. In the infinite models, the universe has always been infinite.
The curvature can be measured (Riess 2007, Kowalski 2008, Komatsu 2010) by multiple methods, to a precision of about 0.6%, and the result is that the error bars currently straddle the line between open and closed cosmologies. We therefore currently don't know whether the universe is spatially finite or infinite. However, the upper bound on the curvature does provide a lower bound on the size of the universe, which is conservatively at least an order of magnitude greater than the size of the currently observable universe.
Komatsu et al., 2010, http://arxiv.org/abs/1001.4538
Kowalski et al., 2008, http://arxiv.org/abs/0804.4142
Riess et al., 2007, http://arxiv.org/abs/astro-ph/0611572
A: It is indeed more useful to cite the age of the universe, because this defines the region in space which is observable, a 13.75 billion lightyear sphere (approximately). 
Clearly, however, the entire universe could be more than 13.75 billion years across in diameter; that number is merely a radius. For example, let's suppose a naive view of the expansion of the universe which doesn't include inflation or dark energy. At the moment of the big bang, photons rush off in every direction at the speed of light (again, naive cosmology - ignore the fact that the universe is opaque for 300,000 years).
These photons are all moving out from one point at the speed of light. We imagine, then, an expanding sphere whose surface is defined by the furthest point which light has so far reached. This sphere is expanding in volume very quickly indeed - the radius is expanding at the speed of light.
So by now, 13.7 billion years later, the radius is 13.7 billion light years. The diameter of the sphere is twice that, 27.4 billion light years. The volume is volume of a sphere with radius r=13.7 GLy, which is $4\pi r^3 / 3 =57.4$ billion cubic lightyears.
This is shows that the universe can easily be much larger than 13.75 billion light years across. Also, note that, if the earth is formed on the expanding sphere, no one from earth will ever be able to "catch up" and see the other side of the sphere, since that side is still expanding. This is what people mean when they say that the universe is larger than we can see.
Now, this answer is wrong. Do not go quoting these numbers. It doesn't take into account inflation or the expansion of the universe. No one knows enough about either of those two effects to give a really good precise number for the size of the universe, but you can be certain that they only result in a bigger universe. One lower-bound for the radius of the universe is 39 billion light years, based on some analysis of the cosmic microwave background.
A: It is impossible to know the true size of the universe.  At best we can make estimates for the age and therefore size of the observable universe.
The reason for this is as mentioned in the original question, the finite speed of light.  If the universe is, as current theories estimate, about 14 billion years old, the most distant point we can see is something that is 14 billion light years away as those are the only objects whose light has had sufficent time to reach us.  There may be things further away but we cannot see them.
It's been a while since I had my cosmology class so any good cosmologists out there can feel free to edit this but the inflationary big bang model of the universe says that at a point very early on, for a very short time, the universe went through an infationary period (hence the name).  During this time, it actually expanded at a rate that is larger than the speed of light.  This means that there are portions of the universe that we cannot see (and never will be able to) since they are forever beyond our light horizon.  So the universe may well be bigger than what we can see but for all intents and purposes, that portion of the universe might as well never exist since we can never interact with it.
The reason this inflationary period doesn't contradict with Special Relativity (at least in my mind) is that it was a change in the very fabric of space itself not thing riding along on that fabric.  The energies of the universe (this occured a very small fraction of a second after the "big bang" long before it was even possible to form matter) were just along for the ride and no information was trying to be exchanged that would have violated the finite speed of light restriction.
A: It's easy to underestimate the size of the Universe because we are concentrating on what we see. What we see when we look at the Cosmic Microwave Background is a look back 13.8 billion years ago to when the Universe was only 1/1000 as large as it is now.
Take a piece of the sky as visualized on maps of the cosmic background radiation (CMB). A piece that is only as large as the Moon, one-half degree across. The entire sky is composed of about 160,000 of such pieces. That piece of Universe has now expanded into an entire volume similar to the one that we are in. Right now, they could look at their CMB map,and see the part of the Universe that we are in as a small part of that CMB map of theirs.
(If we waited 13.8 billion years to watch their part of the Universe expand, we would not see it at all because the expansion will soon cause that area to recede from us much faster than the speed of light!)
So the entire size of the Universe at present, that is represented by what we see on our map of the CMB from 13.8 billion years ago, has about 160,000 times the volume of the currently observable universe, or a radius about 50 times as large, that is about 700 billion light years!
The Universe is actually much larger than that, if not indeed infinite, if we consider those portions that started receding from us faster than the speed of light before the epoch of the CMB.
Note that this is all very approximate, depending on the details of the expansion.
A: Considering that light travels 299 792 458 meter per second while 13.75 billion years is equal to 433 907 732 500 000 000 seconds (13 750 000 000 years x 31 556 926 second a year).
The radius has to be at least: 130 082 265 671 381 485 000 000 000 meter
