Current scientific models say the Universe is still expanding. Are there any theories to explain what the Universe is expanding into?
There are 3 theories as to what will happen next, even though we're not sure which one, if any, is right.
The universe is expanding at an increasing rate, the the three theories as to what is going to happen next are:
"The Big Rip": The universe will expand to the point where it tears the fabric of spacetime, effectively ending the universe.
"The Big Crunch": At some point, the universe will stop expanding and will start compacting, eventually coming together in a single point mirroring the Big Bang. The force of this "Crunch" might start another Big Bang and a new universe, meaning that the universe might have cycles of creation and destruction that come right after each other.
"The Big Chill": The universe will keep expanding forever... and ever... and ever. Once all the stars burn out and there are no more galaxies, all of the heat in the universe will even itself out. To elaborate, by the law of Conservation of Energy, all that heat cannot vanish, so it will spread itself out evenly throughout the universe. Given the ratio of stars to void in the universe, this means that everything will be very, very cold.
Nobody really knows which, if any, of these theories are correct, especially when concerning dark matter and dark energy. They are called "dark" because nobody is really sure what they are or what they do.
Dark matter is called such because scientists found that at the speed many galaxies are spinning, they should be flying apart, scattering stars in every direction. However, some kind of gravity is keeping them together, and scientists dubbed whatever mass is creating that gravity "Dark Matter."
Dark energy is the force theoretically opposing gravity, and is what is making the universe expand at an ever-increasing rate.
Because nobody knows what these are or do, they're kind of wild cards when it comes to predicting an ending. They might have the final, deciding effect on everything, or they might do very little.
tl;dr: Nobody knows for sure!