It is my contention that rebirth or reincarnation is a natural phenomenon that has been a central part of most cultures and tradition, but this fact has for some reason been hidden and attacked for quite a while.
Most of the East has a live and healthy relationship with this to this day, and likewise a lot of people are sharing their own experiences online, there are many Facebook groups with very compelling stories that often checks out when investigated.
Modern research and cases
Ian Stevenson is the best known of the modern researchers, he has documented thousands of cases that often include birth marks from wounds in previous lives, and if I recall correctly also some cases where a reincarnated child points out his own murderer!
The Division of Perceptual Studies at the University of Virginia which he founded in 1967 is still operating, with the reincarnation research led by Jim. B. Tucker M.D.
There are many interviews out there with both of them, here is a recent one: Jim Tucker || The Science of Reincarnation
There are some great, old recordings of a statistician who set up to disprove reincarnation, but all she accomplished was proving it using her tools, I do hope I remember her name soon...
One of the really popular stories the past years has been that of James Leininger
Another one that is very convincing is the story of Jeffrey Keene. His reaction when visiting a Civil War battlefield matched historical event, and also he was able to control a horse at the highest level the first time he ever sat on one :-)
A very recent thing is of course 9/11, there are many cases, this one is the most widely known one.
The Tibetan system
The culture that has the most comprehensive framework for dealing with reincarnation is Tibetan buddhism, within a few kilometers of where I write this in Kathmandu there should be several, maybe a dozen or more Tibetans that are regarded as more or less conscious reincarnations of previous masters.
Having met many of these through the years I have no doubt myself that this is possible, there are so many other abilities that people on this level have.
Here is one interesting documentary on this.
Alexander Berzin has a very good summary of this here
Actually it is not too hard he claims, the problem is to find someone who finds it worthwhile to find your next incarnation! ;-)
The material out there is virtually endless, and it does follow very clear patterns. Its late here, so this will be it for now, will try to continue later!