One possible outcome is that all the matter and energy in the universe will slowly stop moving; the space between the expanding particles will be so great that reaching another particle would take infinite energy. At that point, all energy in the universe will become constant, and the universe will enter a stage of average temperature all over; without any energetic particles attracting or interacting, there can be no consolidation of matter, and so everything in the universe will essentially stop. This is called the Heat Death of the universe, and is a result of Entropy.
Another outcome is that all matter will continue to move apart until gravity can no longer attract; assuming Dark Matter that repulses instead of attracts, the edge of the universe will continue receding until the universe can't hold itself together. All matter will cease to exist as the speed of expansion become all-but infinite. Molecules and atoms will fly apart, and in the end there will be nothing but a superfast cloud of expanding particles, and possible not even those as the atoms explode into quarks and even farther.