Interestingly enough, there are several solutions to Einstein's equations that do allow time travel. Going really quickly doesn't work, but the much, much funner world of General Relativity certainly allows it.

There are a few ways of pulling it off:

1 - Get a long, heavy (The heavier the better) rod, and get it rotating (The faster the better). Travel round and round it. Due to the distortion of space caused by the rotation of the mass, travelling in one direction around the rod 'speeds up' time and travelling in the other sends you backwards in time. If you go fast enough around a heavy enough rod that's rotating fast enough, you just might manage to go back further then the time it took for you to travel around. Maybe. But, basically, this one is unworkable, currently. (By rod, I mean a long, cylindrical object that is longer then a human. It would work with smaller ones, but then the space-time distortion is much, much smaller).

2 - If you can get a beam of light to bend into a circle, somehow (Distortion of space-time to the rescue, again!), and then walk around that circle, the same thing happens. But causing that much distortion takes more energy then option 1. Not happening any time soon.

3 - Make a wormhole. I don't care how, that's irrelevant. Wormholes are most certainly allowed by the current model of physics.

Take the 'in' end of the wormhole, and send it speeding off much, much faster then the other end travels. Due to time dilation, when the 'in' end gets back to the same place as the 'out' end, the 'in' end will be younger. Hop in the 'in' end, and you pop out of the 'out' end... back when it was the same age as the 'in' end was when you hopped into it.

That is probably the most workable solution, but it's still problematic. Current wormholes (Which are essentially mathematical constructs) have an annoying problem - you need to use matter with negative energy density to keep the darn things open, and even then, your gravity causes the damn thing to fall apart as you go through. And it falls apart at the speed of light, so it's impossible to get through them. However, it has recently been proven that very, very strong magnetic fields can have the same effect, creating wormholes that don't require matter with negative energy density, and don't collapse behind you. However, we're talking massive magnetic fields - way beyond anything that can currently be generated by any technique we know of.

There is one major flaw with all of these methods, though - You can't go back in time past the creation of the actual machine.

SSGX, you know how I just listed three methods that let you go backwards?

Yeah.

You just can't go back beyond the initial creation of the time machine. Which does rather nicely deal with the Audience Paradox (We would expect lots of people from the future to want to go to important events from the past - say, the forming of the United Nations, but we know from historical records that there weren't massive quantities of people there. Where are they?)

Actually, 'the numbers add up correctly' generally mean it DOES work. Relativity is a theory - it has been confirmed and tested time and time again. Just because length contraction seems so far 'out there', doesn't mean you dismiss it. It works. The numbers add up. If you don't dismiss length contraction on the basis of it breaking common sense, then don't dismiss time travel on the same grounds. Common sense tends to be wrong. That's the problem with it. Intuitively, if something is walking forwards (at 1 m/s) on a bus that's travelling at 10 m/s, the person's velocity with respect to the ground is 11 m/s. That's common sense.

Now, let's say a spaceship is travelling at 0.8c, and you are moving in an onboard transport at 0.3c. You don't travel at 1.1c - that's impossible. Common sense is wrong. Velocities don't add - the formula is much more complicated. It simplifies down to adding at very small velocities - less then about 0.5c.

And you don't need to move anything in the universe besides yourself (And possibly some machinery).

Say you move 4 metres upwards - 4 metres in the Y-axis. Everything else in the universe needs to move - when you move upwards, everything else goes down! Therefore, movement in the Y-axis is impossible.

We know movement in the Y-axis is possible, because we do it all the time. So the argument is flawed.

It's really quite simple - you're perspective is different, not the whole of reality. Stuff doesn't stop existing when you close your eyes, it doesn't rotate when you turn around.

Why should movement in the t-axis be any different? Your perspective is changing - NOT REALITY.

Then you don't accept relativity, with all it's marvellous acheivements, and more importantly, experimental evidence. Relativity WORKS on the basis of there being a t-axis. It doesn't work without the concept.

Funnily enough, experiments show several of the effects predicted - Gravitational redshift, time dilation, deflection of light by gravity, etc. etc.

But several of the effects of relativity don't work unless you have time as an actual axis - it isn't just a pretty mathematical construct (And you wouldn't believe how real a helluva lot of mathematical constucts have turned out to be)

For example, time dilation. How do you explain that with 'time as a function of space' rather then 'time as an axis'? What about the effects of gravity on the 'speed' of time? (Gravity slows it down. This one is confirmed)