The story follows the crew of 25 men and 25 women of the starship Leonora Christine who are on their way to colonize a planet in orbit around the star Beta Virginis, in the constellation Virgo.
The starship is not capable of faster-than-light travel, but due to relativity and time dilation, the crew expects to spend 5 years on board the starship while 33 years will pass by on Earth. However, the ship passes through a small nebula which damages the "deceleration field generators". The ship cannot slow down......it actually continues to accelerate rapidly. Because of the radiation produced by the engines, the crew cannot repair the damage. The starship is unable to stop and is doomed to travel endlessly through time and space.
The speed increases to the point that billions of years pass by on Earth. So, I suppose technically we can say the crew advances to the future, although relatively little time passes for them.
Tau Zero is classified as a "hard science fiction novel" - the plot is grounded in real physics. Not exactly my favorite type of Sci-Fi. I don't mind a bit of real science, but for that I'd rather read Simon Singh or Stephen Hawking. Too much description of the hard science takes away from the story line.
The situation looks bleak throughout 95% of the story, with a happily-ever-after ending coming in the final chapter. With Sci-Fi, you can always manage a happy ending if you want one.
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