Reincarnation as a religious concept states that a biological sentient being, once it sheds its temporary shell of flesh is reborn into a new biological physical body in a continuous cycle of death and rebirth.
Eastern Religions, Hinduism, Buddhism, Sikhism and Jainism incorporate this doctrine into their belief systems.
It is sometimes referred to as the Saṃsāra or sansara doctrine. The Sanskrit word samsara has no equal in the English language, but has been translated to mean a form of 'wandering' or transmigration with connotations of a cyclical nature. It is an integral part of the 'karma' theory which most are somewhat familiar with.
There are some indications that ancient philosophers in the west had similar concepts. Plato wove reincarnation into several of his allegories. Plato as well as other ancient Greeks believed [or preached] that the soul about to be reincarnated drank from the river of forgetfulness [River Lethe] which was a tale to explain why people had no conscious recollection of past lives.
Ancient post primitive and tribal groups of Europe had aspects of reincarnation and transmigration of the soul within their traditions. Julius Cesar wrote of the druids ...
'The principal point of their doctrine is that the soul does not die and that after death it passes from one body into another... ' 
Some tribal groups among the Africans, Native Americans, Pacific Islanders and Australian Aborigines had belief systems that included reincarnation to varying degrees.
In Judeo- Christian lore and scriptures there is little overt indication of a belief in reincarnation in the classical sense other than the concept of eternal life, heaven and hell yada yada yada. Some scholars have argued that the concept is woven into the scriptures in the 'mystery' tradition only intended for 'those who have ears to hear'.
Scientist Carl Sagan once asked the Dalai Lama what he would do if reincarnation were definitively disproved by science. The Dalai Lama answered, 'If science can disprove reincarnation, Tibetan Buddhism would abandon reincarnation, but it's going to be mighty hard to disprove reincarnation' 
Science can not disprove reincarnation, some pompous scientists have mocked it but adhering to the scientific method and remaining within the parameters of objectivity there is no feasible basis to declare that reincarnation is a false doctrine. In fact there have been many studies seeking to prove or at the very least lend credibility to the doctrine / theory of reincarnation.
One such theory postulated by Dr. Robert Lanza, a specialist in regenerative medicine, stem cell research and known in the scientific realm for several successful experiments in cloning. Lanza later delved into physics and quantum mechanics and devised his theory or treatise of 'Biocentrism'.
Without becoming overly boring with intricate details, Biocentrism basically implies that death of consciousness does not exist. Death exists in our minds because we equate our beings with our physical bodies, therefore if the body perishes so too does the consciousness. 'But if the body receives consciousness in the same way that a cable box receives satellite signals, then of course consciousness does not end at the death of the physical vehicle.
In fact, consciousness exists outside of constraints of time and space. It is able to be anywhere: in the human body and outside of it. In other words, it is non-local in the same sense that quantum objects are non-local. ' 
Quantum consciousness, as theorized by Dr. Lanza helps to illuminate reincarnation without needing to invoke religion or pre-instilled biases. Your consciousness exists outside the physical shell of flesh and recycles into a new body.
Case Studies in reincarnation
Past-life recognition nd the closely related realm of past life regression has stoked inquisitive minds in recent times. Everybody from new world tree huggers to serious theorists and scientific researchers have been studying it. Past-life recognition in children who apparently whom did not drink deeply of Platos 'River Lethe' seems to offer a lucrative reservoir of information.
In the 1960s Dr. Ian Stevenson interviewed thousands of people who claimed to have had past life memories. Stevenson found that Past-life recognition by pre adolescent children was much more common than previously thought.
Dr. Stevenson and an associate from the University of Virginia Medical Center Division of Perceptual Studies, compiled more than 2500 case studies that chronicled Past-life recognition by young children.
The following links are to external articles regarding case studies of purported instances of reincarnation. Mouse over the links for more detailed data without clicking.
1. James Leininger The Past Life Memories of James Leininger
2. Edward Austrian Mum discovers her 4-year-old used to be an 18-year-old soldier named James.
4. Ryan of Oklahoma Return to Life: Extraordinary Cases of Children Who Remember Past Lives Chapter 5
See Also Psychology Today - Children Who Seemingly Remember Past Lives
5. Suzanne Ghanem Old Souls: The Scientific Evidence For Past Lives Pg 80 -87
6. Shanti Devi Old Souls: The Scientific Evidence For Past Lives
7. Barbro Karlen Reincarnation Case of Anne Frank | Barbro Karlen
8. Imad ElawarReincarnation, Birthmarks and the Druze
9. Marta Lorenz - Twenty Cases Suggestive of Reincarnation:
10. Pollock Twins
1. De Bello Gallico Julius Cesar