expanding universe
The expanding universe is general relativity applied to the cosmos as a whole. When Einstein's equations are solved for a universe that looks roughly the same everywhere and in every direction, the solutions, called the FLRW models, do not allow it to sit still: space itself must either stretch or shrink over time. Our universe is stretching, so the distances between far-apart galaxies grow as the years pass, carrying them apart.
The vital subtlety is that the galaxies are not flying outward through space like debris from a bomb. Instead, space between them is expanding, the way ink dots on a balloon's surface drift apart as the balloon inflates, even though no dot moves across the rubber. This is why distant galaxies recede faster the farther away they are, the relationship Edwin Hubble found in 1929: more stretching space lies between us and them, so more of it expands. There is no centre and no edge that the expansion runs away from.
Run this expansion backwards and everything was once packed far closer, hotter, and denser, which is the basis of the Big Bang picture and is confirmed by the faint afterglow of heat that fills the sky. The stretching of space also lengthens the wavelength of light travelling across it, shifting distant galaxies' light toward the red, and it is this cosmological redshift that lets us measure how fast and how long the universe has been growing.
A galaxy's recession speed v rises with its distance d; the constant H_0 sets the current expansion rate.
The galaxies are not racing through space; space between them is growing. Over very large distances this growth of space can carry galaxies apart faster than light without violating relativity, since nothing is moving through space faster than light.