Dr. Ian Stevenson did research on rebirth, and most of his subjects were children.
As I remember he said during the first 5-6 years of a child's life some of them would remember how they died.
Most of them who remembered their previous life have had some sudden tragic death.
Hence, these incidents seem to be deeply carved in their minds, but this memory would slowly fade away as they grew old.
Most of these incidents were reported from families having Hindu or Buddhist backgrounds because their parents believed in rebirth and believed their child was saying the truth.
In some cases, when the researchers traced the child's past routes based on their stories, they had actually found that the person had existed.
Also in some cases it turned out that the parents have been lying to gain some popularity from local media.
:)
But I think this research is not so popular among other scientists, claiming it has no clear evidence that backs his theories.
Upon his death I think his student continues this work.
Here's link to their work.
http://www.medicine.virginia.edu/clinical/departments/psychiatry/sections/cspp/dops/publications-page
(Edited the previous reference according to Ven.
Yuttadhammo's comment)
buddhism.stackexchange.com/users/701/dmsp
First and foremost account is the Jataka stories .
These are generally the account of Buddha himself.
Also there are some records of his disciples.
In modern times a famous person who did research in this area is Edgar Cayce .
Also there has been Parapsychology research in this area.
Keep in mind if it is a monk who has developed Abhijñā , they do not openly say they have such powers.
(I know and seen some monks with these abilities who I have been very close to, but they will not come out to the open and show this so if I way so and so has it there is not way to ascertain my claim.)
As a final note, what I can say is Ehipassiko.
Come and see for yourself.
Develop the 4 Jhanas and see for yourself.
buddhism.stackexchange.com/users/295/Suminda Sirinath S. Dharmasena
Stevenson has published a summary of his work at the end of his life, see Medical Hypotheses (2000) 54(4), 652–659
I add the summary of his paper:
Several disorders or abnormalities observed in medicine and psychology are not explicable (or not fullyexplicable) by genetics and environmental influences, either alone or together.
These include phobias and philiasobserved in early infancy, unusual play in childhood, homosexuality, gender identity disorder, a child’s idea of havingparents other than its own, differences in temperament manifested soon after birth, unusual birthmarks and theircorrespondence with wounds on a deceased person, unusual birth defects, and differences (physical and behavioral)between monozygotic twins.
The hypothesis of previous lives can contribute to the further understanding of thesephenomena.
buddhism.stackexchange.com/users/880/Jo Wehler
Well, I am buddhist.
I am from Sri Lanka.
I told my story of past life when I was 2yrs old.
I have seen many others who have told.
buddhism.stackexchange.com/users/882/nimala9
The famous claim of Bridey Murphy 1923 - 1995 to have previously lived in 19th century
Ireland is in Wikipedia http://en.wikipedia.org/wiki/Bridey_Murphy which has references to variously sceptical reports and evidence.
The book "Have you Lived Before this Life" by L. Ron Hubbard (1968) reports uncritically on 43 cases of remembered previous lives.
It expressly contains no evaluation of the truth or falsity of the data, and each investigation was done as an "auditing" session or scientology confessional, where the investigator prompts the subject to recall usually painful, lurid episodes.
Hubbard himself reported an expedition to verify his own claimed past life memory in "A Test of Whole Track Recall" (1968) republished in 1972 as "Mission Into Time" http://scientology.wikia.com/wiki/Mission_Into_Time .
These books may not be in print now but Knud Eriksen, a Dane who is not an uncritical scientologist, has investigated http://www.solitarytrees.net/racism/pastlife.htm
the claimed past lives of Hubbard - there are several!
- and "finds no good reason to doubt" them.
Buddhists must continue to have doubts about Ron Hubbard's claim in "Hymn of Asia" to have been Buddha (Siddharta Gautama).
buddhism.stackexchange.com/users/959/cuddlyable3
Although the vast majority of people will not remember past lives, are there reliable accounts of people who have? Specifically are there accounts of Buddhists who have remembered past lives and whose experience can be (even partially) verified in some independent way? Or do people with past life memories generally come from other religions and traditions such as Hinduism?  I appreciate that reliable is a very ambiguous term. If possible could people perhaps note the quality of the evidence when answering. Is it anecdotal, statistical significant etc...?
Are there reliable accounts of Buddhists who have remembered past lives?
buddhism.stackexchange.com
<rebirth><science>
buddhism.stackexchange.com/questions/3501
buddhism.stackexchange.com/users/157/Crab Bucket
S2: The book "Have you Lived Before this Life" by L. Ron Hubbard (1968) reports uncritically on 43 cases of remembered previous lives. Hubbard himself reported an expedition to verify his own claimed past life memory in "A Test of Whole Track Recall" (1968) republished in 1972 as "Mission Into Time" http://scientology.wikia.com/wiki/Mission_Into_Time. In addition, there is the famous claim of Bridey Murphy 1923 - 1995 to have previously lived in 19th century. There is anecdotal evidence of people telling stories of past lives at 2 years of age and the famous claim of Bridey Murphy 1923 - 1995 to have previously lived in 19th century. The book "Have you Lived Before this Life" by L. Ron Hubbard (1968) reports uncritically on 43 cases of remembered previous lives. Hubbard himself reported an expedition to verify his own claimed past life memory in "A Test of Whole Track Recall" (1968) republished in 1972 as "Mission Into Time".
Anecdotal evidence of people telling stories of past lives at 2 years of age and the famous claim of Bridey Murphy 1923 - 1995 to have previously lived in 19th century. The book "Have you Lived Before this Life" by L. Ron Hubbard (1968) reports uncritically on 43 cases of remembered previous lives. Hubbard himself reported an expedition to verify his own claimed past life memory in "A Test of Whole Track Recall" (1968) republished in 1972 as "Mission Into Time".