spacetime - According to General Relativity, Does The Past "Exist"?

I'm curious about just what is meant by time being another dimension, like the three (observable) spatial dimensions. Does this imply, according to General Relativity, that the past and the future already "exist" and that we're just moving through it, as we move through spatial dimensions? Or, is time being a dimension just a mathematical construct allowing us to calculate time dilation effects? Do we even know the answer to this question?

Intuitively, it would seem to me that the past doesn't "exist" except in our memories and that the future "hasn't happened yet" so that all that exists is the present...

Clarification

I suppose I should clarify what I mean by "exists". The three spatial dimensions definitely exist. You can travel through them and they are not just a bit of abstract mathematics.

What you can't do is travel faster than C through space. But, if you could you know when you stop, there would be something there.

Now, if you were to take the hypothetical example of time travel, there are two distinct questions here that follow this same line of reasoning. For the present, we don't have the technology to travel through time and it seems like an extremely difficult thing to do. But if you could would there be something there to travel to?

So, even if it is impossible to travel through time, our theories could still inform us whether the past and future exist in the same sense as the present does along the above lines. This was not supposed to be a time travel question... Just a question about the fundamental nature of spacetime: all "presents" existing, or only one "present", and what our current theories tell us or don't tell us about the answer to that question.

Further Clarification

So, as one of the answer's helpfully pointed out, Relativity certainly states there is no "privileged present", or a present that nature prefers and that all other points in time are measured against. I understood this before I asked the question.

But, you could still ask the question of whether in an observer's own frame of reference, does his own past and future exist?

To illustrate: Imagine instead of three spatial dimensions, we had only two. When space would stretch, it would expand or contract a grid drawn on the "paper" universe's spacetime. But, the "paper" would not move out of it's plane (NOT like a trampoline).

Time would be the third dimension, with infinitesimal space sheets stacked one after another. A stretching of time would mean pulling the sheet in or out like a trampoline.

So, in this illustrative universe, if the past and future exist, there are a nearly infinite number of sheets of present slices in array from the universe's birth till it's death. If only the present exists, there is only ever one slice and it changes over time.

Can we say which view is correct in a concrete way, not just in terms of the math? (If you haven't noticed, I'm not a math guy, but I am a picture guy)