If the Universe is expanding and galaxies are rushing away from one another, why are the galaxies in the Local Group not flying apart?
In the modern Universe, gravity is the main (often only!) force shaping the large-scale structure of the universe (galaxy-sized objects, groups of galaxies, and clusters of galaxies). The expansion of the universe wants to push galaxies further apart, but if enough mass is in a small enough area, gravity can stop the expansion of the universe in that region. This is why the planets are not moving away from the sun, why the galaxy doesn't fly apart, and why the Local Group of galaxies (the Milky Way, the Andromeda Galaxy, and several smaller galaxies) stays together.
Another question some people have is whether the “dark energy” of the universe, the “cosmological constant” that may be causing the expansion in the universe to accelerate, will result in our galaxy getting ripped apart in the future. Again, gravity wins, and everything up to our local group of galaxies will remain together in the future. There are larger structures in the Universe bound together by gravity, such as clusters of galaxies, where the local expansion of space has ceased. Superclusters of galaxies are known to exist, and these have enough gravitational pull to eventually overcome the expansion of the universe, but they haven't overcome the expansion yet.
The nearest cluster of galaxies is the Virgo cluster of galaxies. Currently, the Virgo cluster is receding from us at a rate of about 1200 kilometers per second. If there is no cosmological constant, the gravity of the Virgo cluster would stop this expansion, and the Local Group of galaxies would fall into the Virgo Cluster in the very distant future. But with the possible strength of the cosmological constant, it is no longer certain if the gravitational pull of the Virgo Cluster is strong enough to stop the Local Group of galaxies. If this is the case, then in the very distant future, the acceleration of the universe will push all the galaxies except for those in the Local Group beyond our sight, and we'll be left alone in a much duller Universe.