In the Bhagavad Gita, which is a part of the Mahabharata, reincarnation is clearly stated as a natural process of life that has to be followed by any mortal. Krishna says:

Just as the self advances through childhood, youth and old age in its physical body, so it advances to another body after death. The wise person is not confused by this change called death (2,13). Just as the body casts off worn out clothes and puts on new ones, so the infinite, immortal self casts off worn out bodies and enters into new ones (2,22).

In the Puranas the speculation on this subject is more substantial and therefore specific destinies are figured for each kind of "sin" one performs:

The murderer of a brahmin becomes consumptive, the killer of a cow becomes hump-backed and imbecile, the murderer of a virgin becomes leprous - all three born as outcastes. The slayer of a woman and the destroyer of embryos becomes a savage full of diseases; who commits illicit intercourse, a eunuch; who goes with his teacher’s wife, disease-skinned. The eater of flesh becomes very red; the drinker of intoxicants, one with discolored teeth.... Who steals food becomes a rat; who steals grain becomes a locust... perfumes, a muskrat; honey, a gadfly; flesh, a vulture; and salt, an ant.... Who commits unnatural vice becomes a village pig; who consorts with a Sudra woman becomes a bull; who is passionate becomes a lustful horse.... These and other signs and births are seen to be the karma of the embodied, made by themselves in this world. Thus the makers of bad karma, having experienced the tortures of hell, are reborn with the residues of their sins, in these stated forms (Garuda Purana 5).

Similar specific punishments are figured by The Laws of Manu (12, 54-69).

According to the Upanishads and Vedanta philosophy, the entity that reincarnates is the impersonal self (atman). Atman lacks any personal element, reason for which the use of the reflexive pronoun "self" is not quite right. Atman can be defined only through negating any personal attributes. Although it constitutes the existential substrata of man’s existence, atman cannot be the carrier of one’s "spiritual progress", because it cannot record any data produced in the illusory domain of psycho-mental existence. The spiritual progress one accumulates toward realizing the atman-Brahman identity is recorded by karma, or rather by a minimal quantity of karmic debt. According to one’s karma, at (re)birth the whole physical and mental complex man consists of is reconstructed, all that pertains to the world of illusions. At this level, the newly shaped person experiences the fruits of "his" actions from previous lives and has to do his best to stop the vicious cycle avidya-karma-samsara.

As a necessary aid in explaining the reincarnation mechanism, Vedanta adopted the concept of a subtle body (sukshma-sharira), attached to atman as long as its bondage lasts, which actually records the karmic debts and transmits them from one life to another. However, this "subtle body" cannot be a form of preserving one’s personal attributes, as it does not offer any actual data belonging to previous lives to the present conscious psycho-mental life. All this kind of data is erased, so that the facts recorded by the subtle body are a sum of hidden tendencies or impressions (samskara) imprinted by karma. They will materialize unconsciously in the life of the individual, without giving him any hint for understanding his actual condition. There is no possible form of transmitting conscious memory from one life to another, because its domain belongs to the world of illusions and dissolves at death.

In the Samkhya and Yoga darshanas, the entity that reincarnates is purusha, an equivalent of atman. Given the absolute duality stated between purusha and prakriti (substance), nothing that belongs to the psycho-mental life can pass from one life to the other because it belongs to prakriti, which has a mere illusory relation with purusha. However, in the Yoga Sutra (2,12) is defined a similar mechanism of transmitting the effects of karma from one life to another, as was the case in Vedanta. The reservoir of karmas is called karmashaya. It accompanies purusha from one life to another, representing the sum of impressions (samskara) that could not manifest themselves during the limits of a certain life. In no way can it be a kind of conscious memory, a sum of information that the person could consciously use or a nucleus of personhood, because karmashaya has nothing in common with psycho-mental abilities. This deposit of karma merely serves as a mechanism for adjusting the effects of karma in one’s life. It dictates in an impersonal and mechanical manner the new birth (jati), the length of life (ayu) and the experiences that must accompany it (bhoga).

Buddhism denies the reality of a permanent self, together with all things pertaining to the phenomenal world. The appearance of human existence is generated by a mere heap of five aggregates (skandha), which suffer from constant becoming and have a functional cause-effect relation: 1) the body (rupa) - that consists of material form and senses, 2) feeling (vedana) - the taste of any experience, 3) cognition (sanna) - the process of classifying and labeling experiences, 4) mental constructions (sankhara) - the states which initiate action, and 5) consciousness (vijnana) - the awareness of a sensory or mental object. The five elements, as the whole assembly they construct, are impermanent (anitya), undergo constant transformation and have no abiding principle or self. Man usually thinks that he has a self because of consciousness. But being itself in a constant process of becoming and change, consciousness cannot be identified with a self that is supposed to be permanent. Beyond the five aggregates nothing else can be found in the human nature.

However, something has to reincarnate, following the dictates of karma. When asked about the differences between people in the matters of life span, illnesses, wealth, etc., the Buddha taught:

Men have, O young man, deeds as their very own, they are inheritors of deeds, deeds are their matrix, deeds are their kith and kin, and deeds are their support. It is deeds that classify men into high or low status (Majjhima Nikaya 3,202).

If there is no real self, who inherits the deeds and reincarnates? Buddha answered that only karma is passing from one life to another, using the illustration of the light of a candle, which is derived from other candle without having a substance of its own. In the same manner there is rebirth without the transfer of a self from one body to another. The only link from one life to the next is of a causal nature. In the Garland Sutra (10) we read:

According to what deeds are done
Do their resulting consequences come to be;
Yet the doer has no existence:
This is the Buddha’s teaching.

The Tibetan Book of the Dead describes in detail the alleged experiences one has in the intermediary state between two incarnations, suggesting that the deceased keeps some personal attributes. Although it is not clear what actually survives after death in this case, there is mentioned a mental body that cannot be injured by the visions experienced by the deceased:

When it happens that such a vision arises, do not be afraid! Do not feel terror! You have a mental body made of instincts; even if it is killed or dismembered, it cannot die! Since in fact you are a natural form of voidness, anger at being injured is unnecessary! The Yama Lords of Death are but arisen from the natural energy of your own awareness and really lack all substantiality. Voidness cannot injure voidness! (Tibetan Book of the Dead, 12)

Whatever the condition of the deceased after death might be, any hypothetical personal nucleus vanishes right before birth, so there can be no psycho-mental element transmitted from one life to another. The newborn person doesn’t remember anything from previous lives or trips into the realm of intermediary state (bardo).

Another important element in the Buddhist theory of reincarnation is the extreme rarity of being reincarnated as a human person. The Buddha taught in the Chiggala Sutta (Samyutta Nikaya 35,63):

Monks, suppose that this great earth were totally covered with water, and a man were to toss a yoke with a single hole there. A wind from the east would push it west, a wind from the west would push it east. A wind from the north would push it south, a wind from the south would push it north. And suppose a blind sea-turtle were there. It would come to the surface once every one hundred years. Now what do you think: would that blind sea-turtle, coming to the surface once every one hundred years, stick his neck into the yoke with a single hole?
It would be a sheer coincidence, lord, that the blind sea-turtle, coming to the surface once every one hundred years, would stick his neck into the yoke with a single hole.
It's likewise a sheer coincidence that one obtains the human state. It's likewise a sheer coincidence that a Tathagata, worthy and rightly self-awakened, arises in the world.

If one would try to calculate the probability of obtaining the human state according to this text, and consider the surface of  "this great earth" as being just the surface of India, the odds would be one chance in a time span in years of 5 followed by 16 zeros. This is 5 million times the age of the universe.

Reincarnation is a teaching hard to find in the aphorisms of the Tao-te Ching (6th century BC), so it must have appeared later in Taoism. Although it is not specified what reincarnates, something has to pass from one life to another. An important scripture of Taoism, the Chuang Tzu (4th century BC), states:

Birth is not a beginning; death is not an end. There is existence without limitation; there is continuity without a starting point.
Existence without limitation is space. Continuity without a starting point is time. There is birth, there is death, there is issuing forth, there is entering in. That through which one passes in and out without seeing its form, that is the Portal of God (Chuang Tzu 23).

Once the Eastern concept of reincarnation arrived in Europe, its meaning changed. During the Middle Ages it was a doctrine reserved for the initiates of some occult traditions (Hermetism, Catharism, etc.), who have taken it over from Neo-Platonism. A larger acceptance of reincarnation was promoted in the Western world only beginning with the last century, by the efforts of Theosophy, and later Anthroposophy. Their intense ministry, combined with that of many Eastern gurus, and especially the efforts of the New Age movement, determined a wide acceptance of reincarnation in our society today, so that this concept became one of the most fascinating doctrines in explaining the origin and meaning of life.

However, its modern version is substantially different from what Eastern religions stated. Far from being a torment out of which man has to escape by any price through abolishing personhood, New Age thinking considers reincarnation as an eternal progression of the soul toward higher levels of spiritual existence. Influenced by the Christian cultural context but totally opposing Eastern classic ideology, many consider today that the entity that reincarnates is our soul, which preserves the attributes of personhood from one life to another. This compromise obviously emerged from the desire to adopt the reincarnation doctrine to Western thought. The concept of an impersonal atman reincarnating was too abstract to be easily accepted, so Westerners needed a milder version of this doctrine. Although this tendency proves the soul’s yearning for a personal destiny, it doesn’t bear too much resemblance to classical Eastern spirituality, which rejects it as totally perverted.

The above information about the meaning of reincarnation in the Eastern religions and the nature of the entity which is reincarnating will be helpful in examining the modern proofs for it, which are so popular today. While analyzing them, we need to remember that according to the Eastern concept of reincarnation there cannot be any personal element that could wander from one life to the next.