The fabric of space-time is expanding in every direction. All the stars in the night sky are receding from our view: the more distant the star, the faster the retreat. Someday the night sky will be black, as even the closest star (if any still burn on) is racing away faster than the light that brings us its news. This is not a violation of Einstein's prohibition against moving faster than light: that rule only applies to matter and energy, not to space-time itself.
What's going on? Empty space is growing around us. I've written about this before, using the analogy of a universe on a ripple spreading out across a pond. Physicists aren't sure what's causing the expansion of the universe, so they give it the mysterious moniker "dark energy". If there really are mysterious bearers of this force, dark-trons you might call them, they account for 74% of the energy/mass in the universe.
Physicists used to believe in a Big Crunch (some still do, I'm sure) where the Universe gets pulled back together into a point, a reverse Big Bang. This lead naturally to the Big Bounce theory, where immediately after the Big Crunch you've got a new Big Bang. The universe would cycle endlessly (although possibly slowly winding down...), life would begin again and again and again.
I recall wondering about this as a child, and what it meant for humanity's future. It seemed to mark a fixed end to our days. No matter how successful our civilization is, no matter how many star systems we colonize, it would face extinction in the Big Crunch. Sure, a new universe might spring to life, but how could we get there? You can't outrun space shrinking, as there's nowhere else to run to. There could be hope, retreating out of space time for a few million years, but such a task would require entirely unknown laws of physics. From a relativistic standpoint we're doomed in the Big Crunch model.
But then physicists observed that not only is the universe expanding, it's expanding at an ever accelerating rate. It shows no sign of pulling back in for a Big Crunch. So another theory for the end of days took center stage: The Big Freeze. In an expanding Universe, there is increasingly less and less energy per unit volume. Someday a single photon zipping across an empty expanse that once housed our solar system would be an usually warm region of the Universe. Again, not much hope for mankind: this story doesn't end with perpetual rebirths of the Universe we could hitch a ride onto: it ends with order decaying into a vast expanse of nothingness.
So humanity must die. If it comforts you, we're probably talking billions of years of time. And anyways, worrying about humanity's end in terms of the universe's death is a bit like declining desert on the Titanic...but then, there's another theory worth considering...
String Theory has given rise to mathematical models that suggest our universe may not be all there is. In these physical theories, we live on a brane (derived from membrane), a self contained universe floating in a larger reality. Reusing the metaphor from an earlier post, we would be analogous to a civilization living on a ripple in a lake. There are other ripples, there may even be other lakes. Spacetime would be a material we live on, but the energy for the Big Bang would have come from a collision with another sheet of reality.
And suddenly, there's hope again. If space is large enough, it may contain many separate universes, different realities created by different big bangs. These would be unimaginably far away, but if you sit in your spaceship for sufficient aeons you could visit. And even if there's just one, given enough time a new collision will occur: Really, this could happen at any moment. You never know when the space around you is suddenly going to erupt with the energy of trillions of suns. It's amazing how successful physics is at introducing new things for us to worry about (quantum vacuum collapse is a fun one for its combination of utter devastation and quantum weirdness).
So what's this all mean for us? If we can survive sufficiently long in the big freeze, and then survive a universe being born around us, we can keep going as a civilization indefinitely. It seems like a harsh journey for our bodies, but if we programmed the patterns of our DNA into stronger matter it could recreate us once the new universe is born and grown up into a more hospitable place. Ideally nanobots would survive a high energy wave as the new universe passes over them, but if that won't work we could leave patterns of energy to get swept up in the new universe. These would interact at a quantum level with the new matter, so it evolves in a pattern we wished, eventually recreating some simple robot tasked with rebuilding humanity.
As our understanding of physics improves I'll keep you up to date, but the current prognosis is that an eternal civilization is possible (however catastrophically unlikely).