Does anything exist in the intergalactic space? I am a part time physics enthusiast and I seldom wonder about the intergalactic space.  First, it is my perception that all(almost all) the objects in the universe are organized in the forms of galaxies; or in other words, the universe is a collection of billions and billions of galaxies which are spaced far apart.  Does anything actually exist in the empty space between galaxies?  If yes, how did it reach there?
 A: At higher redshifts, we actually see huge clouds of neutral Hydrogen (atomic, not molecular) in the form of absorption lines from distant quasars, called the "Lyman alpha forest"
These clouds may eventually form galaxies and stars, but they are currently just gas, an not particularly low density either. 
A: There certainly is gas within galaxy clusters. It tends to have very high temperature, circa a hundred million K, and is detectable by diffuse x-ray emission. Of course we have the elusive dark matter, which is probably more diffusely spread out than luminous matter. So maybe your question should be modified to, "is the space between galaxy clusters a vacuum?". I suspect the answer to that is also no, although the density is clearly very low.
A: A simple picture is the following: stars are organized in galaxies of various types (like elliptical, spiral, ...). These galaxies are organized in groups which are local aggregates of galaxies (our galaxies is in the so-called "local group" of around 40-50 galaxies). The largest structure is a cluster of galaxies: the groups are bound by gravitational interaction in a cluster.
This classification is not really precise but still. A logical conclusion is that the "empty" space in between these cluster is quasi-completely empty, the density of particles being much much lower than in a galaxy (in average) and is uniform: if it was not uniform, the particles would start to gravitationally interact and start to form other structures.
I think that's a valid reasoning for, say, hydrogen gas (atoms). The conclusion as pointed by @Mark Eichenlaub is then that it can be considered empty and that it is often called "the void".
But of course the space cannot be considered empty: the photons (light) traveling from one place to another are as real as other particles.
The next step in the discussion would involve things like dark energy, but that's a different story.
A: In 1997 the Hubble discovered a large numbers of intergalactic stars. Others have since been discovered. It is now believed that about 1/2 of the stars in the universe may well be rogue stars that are located in intergalactic space. The AVERAGE density of intergalactic space is still very small, however, because of its immense size.
A: Some of the comments and answers suggesting that the space between galaxies is near-empty appear to be somewhat misleading (although, note that some of the answers - like the answers by @Grant Crofton or @Cedric H. - seem to address the space between galaxy clusters rather than between galaxies). Other answers elude to the info below, but here is some clarification about the intergalactic, intracluster space, which from a large-scale viewpoint is very far from empty.
Even without considering dark matter, the majority of matter in galaxy clusters it not in the galaxies, but rather between them in the intracluster medium (ICM), which is a plasma consisting mostly of hydrogen and helium ions (observable in the X-ray portion of the electromagnetic spectrum). In galaxy clusters, around 85% of the mass is in the ICM rather than galaxies / stars. (Then, dark matter makes up even more of the cluster's mass.)
Of course, since galaxy clusters have diameters spanning multiple megaparsecs, the density of the intracluster medium is very low compared to a planet or star. However, the main idea here is that most of the (even baryonic) matter in the universe is actually in the "empty space" between galaxies - so it doesn't make much sense to view that space as "empty" at all.
A: As others have said, it's almost empty, but not quite, as there are gas particles and so on floating around.  As wikipedia states:

Generally free of dust and debris,
  intergalactic space is very close to a
  total vacuum. The space between galaxy
  clusters, called the voids, is
  probably nearly empty. Some theories
  put the average density of the
  Universe as the equivalent of one
  hydrogen atom per cubic meter.
  The density of the universe, however,
  is clearly not uniform; it ranges from
  relatively high density in galaxies
  (including very high density in
  structures within galaxies, such as
  planets, stars, and black holes) to
  conditions in vast voids that have
  much lower density than the universe's
  average.

And that's only if you consider empty to mean void of matter - there's also electromagnetic waves permeating most (all?) of space.  And when you get down to the subatomic level, quantum mechanics ensures that particles are constantly popping into and out of existence as well, even in 'empty' space.
As for how the matter got there, well aside from the normal ways (being shot out of exploding stars and so on), don't forget that before it all started expanding, all of the matter was in the same place anyway, so the particles in intergalactic space haven't necessarily travelled anywhere to get there.  They could have simply stayed where they were while particles around them got gravitationally drawn into nearby clumps of matter/galaxies.
A: I might add some further notes to the actual material things existing in intergalactic space. One might wonder but the notion that there is space is already stating that there is more than nothing.
It implies that there is at least vacuum which is a pretty interesting thing on its own.
Quantum Mechanical harmonic oscillator
Maybe you know that the harmonic oscillator has energy levels
$E_n = \hbar \omega \left( n + \frac{1}{2}\right)$
and an astonishing result is that the lowest energy state is $E_0 = \frac{1}{2}\hbar\omega > 0$.
Quantum electrodynamical oscillator
Coming back to the vacuum, the situation is somewhat comparable. Considering Heisenberg's Principle of Uncertainty in its energy-time form,
$\Delta{t}\cdot\Delta{E} \geq \hbar$
we can see already that a state of a quantum system with definite zero energy for all times cannot exist, even though the expectation value might vanish.
Going more into detail, we see that the operator of the vector potential fullfills the wave equation
$\Delta{A_l} - \frac{1}{c^2}\partial_{tt}A_l = 0$
and a Helmholtz equation if one puts $\partial_{tt}\rightarrow{-\omega^2}$. This equation is usually tackled by separation of variables and after some math we arrive at a Hamilton
$H = \frac{1}{2}\sum_{\lambda}\left({p^2_\lambda+\omega_\lambda^2\lambda{q^2_\lambda}}\right)$
where now $\lambda$ accounts for some mode index. And here comes the magic. This is a description equation for harmonic oscillators! But here we run into a conceptional difficulty. The vacuum energy
$E_{vac} = \frac{1}{2}\sum_\lambda{\hbar\omega_\lambda}$
is infinitely large since there are infinitely many modes of the vacuum. But this is not very physical, so most of the time for calculations you just "leave out" this part.
Implications of a vacuum energy
In the case of different separated domains where you are able to allow a different  different number of modes (e.g. via metal plates), this energy will be different for those domains resulting in a force which is the famous Casimir effect.
But vacuum energy has other implications. One hope it that it might some day explain the cosmological constant in terms of a unified field theory.
So, I hope, I could convince you that "empty" might be much more one would expect :)
Sincerely
Robert
A: http://www.dailygalaxy.com/my_weblog/2017/10/everybody-knew-it-had-to-be-there-missing-half-of-normal-matter-in-the-universe-has-been-detected.html
The matter, baryonic not dark, has mostly been found.
A: Try this link for a semipopular view of the overall picture.
http://www.nasa.gov/multimedia/videogallery/index.html
A: Let us look at numbers.  Exponential numbers.  The "luminous" matter in the universe (what is visible) comprises a volume that can be calculated.  I've come up with a (very) rough total (choosing the cubic kilometer) of 4x10 to the 40th power.  Now calculate the volume of the observed universe - again roughly, 10 to the 70th power.  In this scenario the volume of interstellar and intergalactic space is 10 to the 30th power greater than the volume of observed luminous objects.  That is one million-million-million-million-million times larger.  If the density of that "empty" space is actually 10 to the -29th power of the luminous objects, that would still leave 10 times more matter in the void than in the galaxies, and would account for what is referred to as "dark matter" and its overall gravitational effect.  It is dark because it does not emit light, but IT IS STILL THERE!  It no doubt ranges from subatomic particles to enormous chunks of non-luminous matter.  How can a scientist assume that because we cannot "see" something we can assume it is not there?  It seems obvious to me that Dark Matter is simply the residue left after the formation of galaxies, and that it is far too sparsely scattered to emit light, and even sparse enough to allow the passage of light over vast distances with little aberrational effect. I can't believe I said all that at once.  Sheesh
