Faster Than Light
Conclusion
It might be thought that special relativity provides a short negative answer to this question.  In actual fact, there are many trivial ways in which things can be going faster than light (FTL) in a sense, and there may be other more genuine possibilities.  On the other hand, there are also good reasons to believe that real FTL travel and communication will always be unachievable.  This article is not a full answer to the question (which no doubt will continue to be discussed in the newsgroups for the foreseeable future), but it does cover some of the more common points that are repeatedly made.
It is sometimes objected that "they said no-one would ever go faster than sound and they were wrong.  Now they say no-one will ever go faster than light..." Actually it is probably not true that anybody said it was impossible to go faster than sound.  It was known that rifle bullets go faster than sound long before an aircraft did.  The truth is that some engineers once said that controlled flight faster than sound might be impossible, and they were wrong about that.  FTL travel is a very different matter.  It was inevitable that someone would one day succeed in flying faster than sound, once technology got around the problems.  It is not inevitable that one day technology will enable us to go faster than light.  Relativity has a lot to say about this.  If FTL travel or FTL communication were possible, then causality would probably be violated and some very strange situations would arise.
First we will cover the trivial ways in which things can go FTL.  These points are mentioned not because they are interesting, but because they come up time and time again when FTL is being discussed, and so they are necessary to deal with.  Then we will think about what we mean by non-trivial FTL travel/communication and examine some of the arguments against it.  Finally, we will look at some of the more serious proposals for real FTL.  Many of these things are discussed in more detail elsewhere in the FAQ and hyper-links are provided.  The sections are numbered so that they can be referred to individually.
Trivial FTL Travel
One way to go faster than light is to make the light slow down!  Light in vacuum travels at a speed c which is a universal constant (see the FAQ entry
Is the speed of light constant? ), but in a dense medium such as water or glass, light slows down to c/n where n is the refractive index of the medium (1.0003 for air, 1.4 for water).  It is certainly possible for particles to travel through air or water at faster than the speed of light in the medium, and Cherenkov radiation is produced as a result.  See the FAQ entry Is there an equivalent of the sonic boom for light? .
When we discuss moving faster than the speed of light, we are really talking about exceeding the speed of light in vacuum c (299,792,458 m/s).  The Cherenkov effect is thus not considered to be a real example of FTL travel.
d2X     d2X       p --- - Y ---  = 0         dt2 dx2
Plane wave solutions travel at the speed of sound s where s2 = Y/p.  This wave equation does not allow any causal effect to propagate faster than s.  Relativity therefore imposes a limit on elasticity: Y < pc2.  In practice, no known material comes anywhere near this limit.  Note that even if the speed of sound is near c, the matter does not necessarily move at relativistic speeds.  But how can we know that no material can possibly exceed this limit?  The answer is that all materials are made of particles whose interaction are governed by the standard model of particle physics, and no influence faster than light can propagate in that model (see the section on Quantum Field Theory below).
So although there is no such thing as a rigid body, there is such a thing as rigid body motion; but this is another example in the same category as the shadows and light spots described above which do not give FTL communication.  (See also the FAQ articles The Superluminal Scissors and The Rigid Rotating Disk in Relativity ).
This has solutions of the form:
u = A cos( ax - bt )     c2 a2 - b2 + w2 = 0
These solutions are sine waves propagating with a speed
v = b/a = sqrt(c2 + (w/a)2)
But this is faster than light, so is this the equation for a tachyon field?  (See the paragraph on tachyons below ).  No, it is the usual relativistic equation for an ordinary particle with mass!
Superluminal speeds such as this present no problem once we recognise three types of speed associated with wave motion: phase velocity, group velocity, and signal velocity.  Phase velocity is the velocity of waves that have well-defined wavelengths, and it often varies as a function of this wavelength.  We can combine ("superpose") waves of different wavelengths to build a wave packet, a blob of some specified extent over which the wave disturbance is not small.  This packet does not have a well-defined wavelength, and because it usually spreads out as it travels, it doesn't have a well-defined velocity either; but it does have representative velocity, and this is called its group velocity, which will usually be less than c.  Each of the packet's constituent wave trains travels with its own individual phase velocity, which in some instances will be greater than c.  But it is only possible to send information with such a wave packet at the group velocity (the velocity of the blob), so the phase velocity is yet another example of a speed faster than light that cannot carry a message.
In some situations, we can build a fairly exotic wave packet whose group velocity is greater than c.  Does this then constitute an example of information being sent at a speed faster than light?  It turns out that for these packets, information does not travel at the group velocity; instead, it travels at the signal velocity, which has to do with the time of arrival of the initial rise of the wave front as it reaches its destination.  You might not now be surprised to learn that the signal velocity turns out always to be less than c.
E = mc2/sqrt(1 - v2/c2)
As the speed approaches the speed of light, the particle's energy approaches infinity.  Hence it should be impossible to accelerate an object with rest mass to the speed of light; also, particles with zero rest mass must always move at exactly the speed of light, since otherwise they would have no energy.  This is sometimes called the "light speed barrier", but it is very different from the "sound speed barrier".  As an aircraft approaches the speed of sound it starts to feel pressure waves which indicate that it is moving close to the speed of sound, and before the existence and effects of these waves were well understood, they destroyed several aircraft in the mid 20th century; hence the old name of sound "barrier".  In fact, with more thrust and the right aerodynamics, an aircraft can certainly pass through the sound barrier.
The situation is different for light.  As the light speed barrier is approached (in a perfect vacuum) there are no such waves according to relativity (destructive or otherwise).  Moving at 0.999c is just like standing still with everything rushing past you at −0.999c.  Particles are routinely pushed to these speeds and beyond in accelerators, so the theory is well established.  Trying to attain the speed of light in this way is a matter of chasing something that is forever just out of your reach.
This explains why it is not possible to exceed the speed of light by ordinary mechanical means.  However, it does not in itself rule out FTL travel.  It is really just one way in which things cannot be made to go faster than light, rather than a proof that there is no way to do so.  Particles are known to decay instantly into other particles which fly off at high speed.  It is not necessary to think in terms of the particles' having been accelerated, so how could we say that they could not go faster than light?  What about the possibility of particles that might always have been moving faster than light, and which might be used to send information if they can be detected without ever slowing down to less than the speed of light?  Even if such "tachyons" don't exist (and we don't believe that they do exist), there may be ways of moving matter from A to B faster than light is able to travel from A to B by the usual route, but without anything having to go at a FTL speed locally.  See the paragraph on tachyons below .
Time Travel for more on this subject.
Open Possibilities for FTL Travel
In this last section I give a few of the speculative but serious suggestions for possible faster-than-light travel.  These are not the kinds of thing usually included in the FAQ because they raise more questions than answers.  They are included merely to make the point that serious research is being done in this direction.  Only a brief introduction to each topic is given; more information can be found all over the Internet (and should, like almost everything on the Internet, be taken with a huge grain of salt!).
To begin with, it is rather difficult to define exactly what is really meant by FTL   travel and FTL communication.  Many things such as shadows can go FTL, but not in a   useful way that can carry information.
There are several serious possibilities for real FTL which have been proposed in the   scientific literature, but these always come with technical difficulties.
The Heisenberg Uncertainty Principle tends to stop the use of apparent FTL quantum   effects for sending information or matter.
In general relativity there are potential means of FTL travel, but they may be   impossible to make work.  It is thought highly unlikely that engineers will be   building space ships with FTL drives in the foreseeable future, if ever, but it is   curious that theoretical physics as we presently understand it seems to leave the door   open to the possibility.
FTL travel of the sort science fiction writers would like is almost certainly   impossible.  For physicists the interesting question is "why is it impossible and   what can we learn from that?"