Many people believe that the paradox problem is itself enough to render time travel impossible. Regardless of the physics, the philosophy of time travel is a fascinating area for "thought experiment".
Various paradoxes have been raised, these usually involve an apparent breach of causality in one way or another. The most well-known version is the "grandfather paradox".
The Grandfather Paradox
This is a very simple idea with serious repurcussions for the concept of time travel. Let's say that I invent a time machine and travel into the past. I meet my own grandfather when he was a boy and kill him.
The result? One of my parents is never born, therefore I can never be born.
So I couldn't have gone back in time and killed him.
This is a logical contradiction - and a philosophical nightmare!
Resolutions
There have been numerous proposals for dealing with the apparent causal paradoxes of time travel.
The easiest is simply to say "so what?". The paradox only exists because of our "common sense" view of linear causality - possibly related to the arrow of time. If we step back and look at the system as a whole then we can see a multi-dimensional causality. Unfortunately most of our existing laws of physics assume linear causality in one direction or another, so this resolution is unpopular.
An alternative is to call on the "multiple universes" theory. By travelling into the "past" we are actualy travelling into an alternate or parallel universe. From the moment we arrive, the universes start diverging. Whether the universes "split" or whether they always existed in parallel in some higher "dimension" is a matter of taste.
One side-effect of this resolution is that we can never return to our original time - we are stuck in the parallel universe and can only move forwards into its future. If I shoot my grandfather then get back into my time machine and return to 2013 it will be a 2006 where I was never born. Once I have stepped on that butterfly then my original universe is forever inaccessible to me.
Another interesting idea is the "mobius strip" timeline, discussed here by Anthony Edwards. How the mobius strip generalises to multiple dimensions when multiple time travellers are involved is beyond my grasp - I get lost after the klein bottle!
An interesting technical approach to resolving the paradox is the Novikov self-consistency principle proposed by Dr. Igor Novikov. This essentially says that paradoxes won't happen - it's impossible to create a paradox however hard you try. In this view the universe is in some way "self-righting". If you attempt to shoot your grandfather then something will go wrong - you'll miss, the gun will jam, etc. Or, if you succeed, you'll later learn that your father was adopted; so he still gets born and still marries your mother. This reminds me of the anthropic principle: the universe is this way because if it wasn't we wouldn't be here.
Perhaps the simplest resolution of temporal paradoxes is to say that since they only occur when we travel into the past, then travel into the past is impossible but travel into the future remains a possibility.
Unfortunately this makes time travel rather pointless - it achieves little more than we could get through long-term suspended animation or near light speed travel.
