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Saying “God bless you” or “Gesundheit” in response to someone’s sneezing is extremely common throughout Europe and America. “Gesundheit” simply means “health”, and a post-sneeze wish for good health is found even in the writings of Pliny the Elder.
This early mention of the practice disproves the widespread belief that “God bless you” originated during the 17th Century Plague, when sneezing was supposedly a symptom. The earliest reference in England dates to 1483; also well before the Great Plague.
Superstitions around sneezing probably originated with the earliest people, who may have thought this little explosion in the head was some sort of sign from the gods. Some people still believe that sneezing leads to a temporary deprivation of the soul, with “God bless you” returning the soul to the body.
Reference: Most of the material from this post was found in David Pickering's Dictionary of Superstitions and Steve Roud's The Penguin guide to the superstitions of Britain and Ireland.