A few months ago, I remember reading a book about information theory and how it ties together with many other fields of science. I do not remember the title, but this question had an interesting part in the book. I'm not a real astronomer, but I hope what I think can help. The following is a bit that I remember from the book and a bit of my own thinking:
Let us say there are three levels of our whole "universe". There is the universe, the multi-verse, and the omni-verse. There is a progression involved here. We live in a universe that is constantly in superposition. It is first assumed that quantum physics is true and there is superposition. Each action taken can either be this or that until it is actually perceived. An example of superposition is
Schrödinger's cat. Once the superposition is broken, we take either the A route or the B route. This makes our universe divide up into an A universe or a B universe in which we live in one of them. The collection of all these parallel universes is the multi-verse. Everything that exists would be the omni-verse, and it can be said that it is all of the multi-verses combined. I believe we are testing if there is an omni-verse or not because the end should be the edge of the omni-verse.
I forgot how the book proved that the universe was finite, but I would present my own ideas as there being the Laws of the Conservation of Matter and Energy. If we have the Conservation in place, there must be a finite amount of mass and energy in the universe. Thus, we have a finite universe. Otherwise, an infinite universe would mean these Laws of Conservation didn't matter. We solve the initial question by concluding that there is an end to the universe. There will be a point where we leave our known universe because there will be no more atoms from our universe there. I want to look further into the possibility of an omni-verse because we have the situation of other universes to settle.
The universe constantly divides because superposition is destroyed. That would mean the universes become separate. Since there is only a finite amount of matter, it is possible to find the edge of our universe. We then stumble upon another universe created from superposition. In this universe, though, you might see yourself, but it won't be the same person because the universe is not a sphere in this case. You can't go around the universe and arrive at your initial starting point.
To see if there is an edge to the omni-verse, we need to settle the question of the amount of universes there are. A universe has a finite amount of matter. Let us say that a universe has "x" atoms. All the parallel universes are caused by different combinations of these "x" atoms, yet the amount of atoms is constant. We can go through all the universes with these different combinations, but we will eventually find out that there are no more combinations if we go far enough. Where there are no more combinations of the atoms of a universe, there lies the edge of the omni-verse. That is the point where there is nothing left.
From this, we can see that there is an end to everything, including our universe. This really shouldn't matter because we will probably not reach that far away in our lifetime.
I think a lot of people are misunderstanding the question, thinking if there is an end to the universe time-wise. Please read the full original post before answering the question. Remember that the original author is asking if there is a PHYSICAL end to the universe, not a TIME that everything ends. Stay on topic, everybody.