Julian Baggini in Prospect:
Parfit never became a well-known public intellectual, but within English-speaking academe he is acknowledged as one of the most important philosophers of the late 20th century. He made his name with a single journal paper that breathed new life into an old problem that had drifted into obscurity, mainly because no one had anything new to say about it. The problem was: what needs to be true to correctly identify a person as the same person at two different times?
One obvious answer is something like: do a DNA test. But if having the same body is what makes us the same person, then it doesn’t make sense to conceive of life after death or of uploading ourselves into an AI world, as transhumanists look forward to doing. Nor does it account for the feeling we have that we are, in important senses, not the same person as our toddler selves, or that people in late stages of dementia are not the people they once were.
More here.