There are two very interesting theories on the end of the universe:

1: Heat Death. If the Hubble Constant (the rate at which the universe is expanding) is greater than the universal gravitation constant (the accelleration force that acts on the universe to slow expansion and draw itself back into it's center), then the universe will expan forever. This theory is also called an open universe theory, because it means that the universe is free to exist forever. As the universe expands, eventually all the hydrogen, helium, and other light elements will be used up as fuel for the last remaining stars. When all nuclear fusion of stars and other celestial objects stops, the universe will exist at only a few billionths of a degree Kelvin. Then (per Stephen Hawking's theories), the gluon bonds in protons and nutrons will break down, leaving the universe a sea of quarks. Afterward, the quarks would break down into tiny ammounts of energy, and the universe would exist for eternity filled with zero matter close to, if not at, Absolute Zero.

2: The Giant Crunch. If the aforementioned Hubble Constant is less than the universal gravitation constant, the universe will eventually stop expanding, and begin to contract. The universe would compress into one point in three-dimensional space. Einstein stated that matter and energy cannot be created or destroyed, so when the universe converges on itself, it will have exactally the same properties as it did when the universe began (the same ammount of matter and energy, though not in the same proportions because matter and energy can be changed from one form to the other.) Then, if all goes right, the universe will start over. This theory is called a closed universe theory because the universe can only expand to a certain size before compressing.

Either way, I won't be around to see it. If it's a choice between living in a universe at 0 Kelvin, living in a universe that occupies one 3-d point, or dying, I'd chose dying any day.
There are a great number of schools of thought on how the universe will ultimately "meet its maker" (if you will excuse the euphemism). Here are a few of the more significant ones.

The most classical and often accepted is that it will ultimately experience "heat death." This may be the case if we are in an expanding universe or in a universe where, in what would be basically an infinite amount of time, the Universe would stop expanding. According to the second law of thermodynamics, heat cannot flow from a cold object into a hot one without causing more entropy. Someday, all of the energy will be used up, the universe will be plunged into complete and utter chaos, and therefore there will be no stored up energy left in the entire universe. That means that nobody can do any work (a good thing). It also means that all movement everywhere will cease (a bad thing).

It will never end. Ever. To elude the first and third possibilities, the universe would have to have an infinite amount of reverse entropy stuffs laying around.

The third is the most interesting to me: if the Universe has enough mass to stop from the Big Bang, it will then contract. This means that a kind of reverse entropy will occur: things will go from chaos into order. Broken things will reform! Time will reverse! Cats and dogs living together, the universe as we know it will end!

(slightly updated, embarrassingly soon, due to things pointed out by ariels. Thanks for setting me straight))
With regard to the second "possibility":
It will never end. Ever. This has some amusing ramifications, including eternal recurrence. This would be the case if the universe were to somehow last for an inifinte amount of time. As demonstrated by Henri Poincare, if the particles in the universe are given an infinite amount of time, they can take up an inifnite number of configurations, and therefore entropy will sometimes, in the very far future, be decreased.
This is not what Poincaré demonstrated. Poincaré showed that if your phase space either is compact or has finite volume, then paths are strongly recurrent. This indeed would mean that behaviour would be almost cyclic.

But for the universe "never to end ever" in the sense of simultaneously avoiding a big crunch and heat death, it must have an infinite amount of negentropy to play around with. In particular, you can forget about finite volume conditions -- this would require us to observe finite volumes with infinite negentropy in them, which we certainly do not. And compactness conditions are much the same, for any remotely reasonable geometry.

Infinite negentropy requires an infinite universe, which is not absurd. But then Poincaré's results don't apply.

Revelation 21:1 And I saw a new heaven and a new earth: for the first heaven and the first earth were passed away; and there was no more sea.

Not that I actually believe that - I'm with the heat death theory here.

It seems I need to justify this WU a little more. Until relatively recently, there was very little understanding of physical constraints such as entropy. The question of the end of the universe was a theological one, and opinion was divided (sometimes even within the same religion) as to where the world was going. Hinduism has maintained that the present world will end, but that an endless cycle of worlds will succeed it. The Jewish view has been split between eternal permanence and a final day with a general resurrection. This was a point of contention in Jesus' day, with the Sadducees taking the former view and the Pharisees the latter. Christianity has mostly jumped with the Pharisees, and Revelation reinforces this.

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