If galaxies are moving, why haven't any galaxies collided with our galaxy? 

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While it's true that space is expanding, this is a grossly oversimplified representation of the phenomenon.

We don't yet fully understand the exact reason and mechanics of how the universe is expanding, or how it will behave in the distant future. However, what we do know is that this expansion rate is not uniformly distributed throughout space and matter; that is to say, individual atoms are not getting farther apart from each other. Why would galaxies grow more distant from one another, yet retain their internal structure and composition? The answer is that other forces take predominance over the expansion.

This isn't an unfamiliar principle in physics; many forces have ranges within which they are more or less powerful. For example, the strong force is more powerful than electromagnetism, but it can only act over a short range. The current model appears to show that the forces driving the expansion of the universe are either less powerful, or operating on a different range, than gravity.

We are part of the Virgo Supercluster, a grouping of thousands of galaxies, all of which are interacting through gravity. While the distance between the VC and other superclusters is expanding, all of the contents of the VC are being predominately attracted to each other. Thus, it is entirely possible for galactic collisions to occur, regardless of the expansion of the universe.

The answer to the original question is, frankly, luck. Modeling galactic interactions based on gravity takes place on a scale of millions of years and does not necessarily yield observable results; as Saumya-Dave stated, our current models anticipate the Milky Way and Andromeda galaxies colliding in several billion years. So, in fact, the question is not "why haven't we collided" but "why haven't they collided YET".

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