Dr. Ian Stevenson is a psychiatrist who studies the very strongest cases of young children remembering details about their past lives. Most of these children live in India, and some live in the Middle East. Such a child might start telling her parents, as soon as she can talk, about her "other family", giving details such as her previous name, the name of the village, how she died, and the names of her parents, brothers and sisters, aunts, etc.
These children also act and have the emotions that would fit with the past-life person. She may feel herself to be of a different religion or caste, or to be richer or poorer. She may insist on being taken back to her former family, even if she is punished for saying so.
More than this, such children may actually be able to speak and even answer questions in the language the former person spoke (called "xenoglossy"), or may remember songs or dances from the previous lifetime.
Sometimes, word will get around about the child's statements, and someone in another village will recognize the story about the past-life person. The past-life family is contacted, and arrangements are made for a visit. Most of the time Dr. Stevenson has had to interview the families after this meeting happens, but a few times he has been able to be there when the meeting took place. Dr. Stevenson always takes very careful notes and has other people observing and taking notes as well, to safeguard against him making mistakes and to make sure the reporting is accurate and fair.
In a large number of these cases (at least 250 very strong ones, and thousands of lesser cases), the child will make as many as 40 specific statements about her past life, and most of these turn out to be true. She will point out her former relatives even though people try to trick her; and she will act toward them they way you would expect, and have the emotions you would expect (including both liking and disliking someone). She may give personal, intimate details that only the past-life person could possibly have known, like where a container of gold was buried in the house, or a secret she had told only her husband.
Another very convincing proof of reincarnation Dr. Stevenson has researched has to do with physical birthmarks. Some of the children who have memories of a past life have strange birthmarks or physical deformities, which seem to come from the way the person died in the previous life. In a few cases Dr. Stevenson has been able to look up the autopsy records of the previous person, and match up the photographs showing how they died, with the child's birthmarks. In several cases, a child remembered being shot and killed in a previous life, and remembered enough details that Dr. Stevenson could track down the autopsy records for the past-life person. The autopsy records showed where the bullet entered and exited--and the child had little round birthmarks matching the autopsy.
In a similar case, a boy who had past-life memories had a splotchy birthmark on his chest that looked like scars--when he gave the name of the past-life person, Dr. Stevenson tracked down the name, found the autopsy report, and saw that the previous-life person had been killed by a shotgun blast to the chest. The pattern of the holes in the autopsy looked very much like the boy's birthmarks.