A discordant redshift is a discrepancy between the redshifts of galaxies that lie close together in the sky and that appear to be interacting or to be connected by luminous bridges. The nature of the discordant redshift of members of compact groups of galaxies, such as Stephan's Quintet and Seyfert's Sextet, has been a subject of debate for many years. If the frequency of discordant galaxies is greater than the statistics of chance projection allow, or if apparent connections between discordant galaxies are genuine bridges of matter, as has been argued by Halton Arp, the Burbidges, and others, it challenges the almost universally-held belief that galactic redshifts are due to cosmic expansion and might signify the need for new physical theories. However, although the situation has not been completely settled, evidence now strongly suggests that discordant galaxies do not occur more often than can be explained by line-of-sight coincidences; also, there is independent physical evidence that discordant galaxies all have physical properties consistent with a cosmological distance; for example, those with higher redshift tend to be smaller and fainter than other members of the group.