After Edwin Hubble published his astronomical observations of the redshifted light from nearby galaxies in 1929 it has almost universally been taught that the Universe is expanding. Hubble observed in the light from most of those galaxies that the spectral lines were shifted towards the red end of the spectrum as compared to a local laboratory source of the same atomic gas species. See illustration below. From this he interpreted that it was a Doppler effect, where the galaxies were receding from us the observer.
The Doppler effect is what you experience when you hear the pitch of the sound coming from a speeding train rushing past you. Also you might have had experience with ultra-sound imagining–there the Doppler effect is used to see the flow rate of blood in your arteries and veins. Those examples employ sound waves but the Doppler effect is real physics applied to electromagnetic waves in police radar or weather radar.
The question needs to be asked though, is this the same effect seen in the light from galaxies around us? Cosmologists today say, No! They interpret the observations differently. They say it results from the expansion of space–called cosmological expansion. The galaxies are actually not moving (except for local motion within their own clusters). The universe is expanding and over time it appears to us that they are moving away from us. Hubble discovered, and it has been confirmed many times since, that the greater the distance to a galaxy the greater the spectral lines are redshifted. This is called the Hubble Law.