Allegedly, it was the Romans who began making inscriptions upon tombstones and graves - for them, it was a way of burying, along with the corpse, his/her virtues (which back then were considered to be the personal vanities and values worth of being copied by others). It was also a great way to register a person's deeds, since fame was directly associated with virtue, according to the ancient historian Tacitus.
Romans would even discuss their future epitaphs while having fun with their friends! Although they were very supersticious people, it wasn't considered a bad omen to talk about this kind of subject during parties. And there were all kinds of inscriptions: some recomended people to enjoy life as much as they could... While others cursed their enemies, wishing them several plagues from hell. Nice.