The currently accepted theory about the Universe is that it is expanding from the central point of the Big Bang. According to this model, the matter and energy will continue to expand forever, or until the ultimate state of Entropy exists -- that is, until all matter and energy in the universe comes to a stable point of heat and movement, the point where everything just stops because all the inertia is lost and nothing is generating any new energy.
However, there are multiple theories about the eventual fate of the Universe. Several involve the idea that matter and energy will somehow reverse their expansion and start to contract; this is called the Big Crunch and theorizes that everything in the universe will collapse into a single point, which might then explode in a new Big Bang. Other theories involve the still-uncertain nature of Dark Energy/Matter, which may or may not have an effect on all the other material, and might cause the eventual dissociation of all atoms everywhere.
In every scenario, the ultimate fate of the Human Race is to spread out among the stars, or become extinct in several billion years when the Sun begins its growth into a Red Giant. Humanity can outrace entropy if given enough time to develop close-to/faster-than-light spacecraft, but these innovations are far in the future.