How can a spiral galaxy exist? A spiral arm orbiting a central mass should be dispersed quite quickly as the outer elements would move more slowly than the inner ones. The Milky Way, is about 59 Galactic Years old, which, one would have thought, would be enough rotations to disperse a spiral structure entirely.
Is there, then, something keeping the spiral arms in existence, and if so what could it be? Or are the spiral galaxies monstrous co-incidences?
 A: The material (gas and stars) in the outer part of a galaxy move with roughly the same velocity as the inner part (for example, see this paper), which means that the inner portions do indeed have a faster angular speed; this is sometimes referred to as the "winding problem."
One important feature of spiral arms is that they are bright more because they have lots of young stars than because they have extra material. Young populations of stars include bright, short-lived, blue stars, which die off over time, leaving the fainter, redder populations. Populations of these young stars are particularly apparent in images like this one. Because of this population of bright, young stars, the density of matter in spiral arms compared to the non-arm disk is not as great as their brightness would suggest.
The usual explanation for spiral arms is that they are the result of density waves rather than moving structures. See this paper for a short review.
A: Because of "Dark Matter", and it's everywhere! Astronomers have found that supermassive black holes could live in the heart of galaxies and pull stars at incredible speeds, but it's not strong enough to hold all the stars in the gigantic galaxies together. 
So, to your question of what does hold them together:
A Swiss astronomer named Fritz Zwicky wondered why galaxies stay together in groups; he concluded that there must be something going undetected and coined it to be "Dark Matter." To prove it scientists would build virtual galaxies using computers, simulating stars, gravity and so forth - astronomers were looking forward to such experiments yielding the integrated galaxy similar to our own. But, what they found was disintegration!
The gravity of the galaxy is not enough to hold it together, so scientists added virtual gravity from virtual dark matter to the simulation experiment, and it solved the problem: gravity from dark matter held galaxies together. So, we can say that "Dark Matter is The "Master Of The Universe".
