Current measurements indicate that the expansion of the Universe is accelerating, with Dark Energy pushing the galaxies and clusters of galaxies futher apart. If this continues, then the Universe will end up a very boring place. For many billions of years, the Universe will probably continue much as it is at the moment, with stars forming and dying, burning the lighter elements into heavier ones.
Eventually, many trillions of years in the future, the stars in the Universe will have exhausted their fuel, and the remaining gas won't be dense enough to be able to form more stars. With the clusters being further apart, galaxies won't crash into each other as often, so there won't be as many chances to churn things up and form new stars.
All the matter in the Universe will be condensed into a smaller number of very larger galaxy clusters, with most of the smaller galaxies having been consumed by the larger ones.
From a human perspective, the Sun will die in about another 5 billion years, and the Earth will probably be uninhabitable by then. But who knows where we'll be living by then...
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