In Sanskrit the term antyesti refers to the final sacrifice, the last of the 16 samskaras or life sacraments that mark important events in an individual's life. The antyesti ceremony is the funeral ceremony. This samskara is performed to dispose of the dead body, to give peace to the departed soul, and to enable it to enter the world of the ancestors (pitrs). From the earliest Vedic times cremation was the most common means of disposing of a body. There is, however, written evidence that burial and post burial ceremonies also occurred during the Vedic period. The Rg and Atharva Vedas mention both burial and cremation as legitimate methods for the disposal of the dead.[1] We find evidence in the Aranyakas that the burial of incinerated bones and ashes was an important and elaborate ceremony. By the Grhya and Puranic periods, however, burial and post cremation burial are hardly mentioned. Cremation had become the only orthodox method for the disposal of the dead.[2]

Here is a summary of what we know about cremation from the Rg-veda:[3]

1.  The fire deity, Agni, was invoked to carry the departing soul to the realm of Yama, the god of death.
2. In the case of a priest his sacrificial implements were burned along with his body.
3. Prayers were recited to various deities in order to transfer the departing soul to the world of the pitrs.
4. A cow or goat, known as an anustarani[4], was burned along with the body of the deceased.
5. In the case of a deceased husband, the wife would lay on the funeral pyre along side the body of her husband. Before the fire was lighted, she would be asked to rise from the side of her husband's body and rejoin the living.[5]

The Atharva-veda (XVIII) adds the following information:

1. The body was dressed in new garments before cremation.
2. Grains and sesame seeds were scattered along side the body before cremation.
3. The pitrs were ritually invoked to attend the ceremony and invited to sit on the southern side of the fire.
4. Streams of ghee along with prayers were offered to the pitrs during the cremation.
5. Prayers and oblations made of rice cakes, milk, meat, whey, honey, and water were used in the worship of various gods in order to ensure long life and prosperity for the living relatives.
6. Prayers and oblations were offered to three generations of pitrs: the father, the grandfather, and the great grandfather, during the cremation.
7. Cakes of rice, sesame and other articles of food were buried along with the cremated bones.

It is evident from the Atharva-veda that the worship of pitrs had its origins in the earliest Vedic period.

The cremation process during the Grhya period may be summarized as follows:

1. As soon as the person died a cremation pit called a smasana was dug. The pit was made in a fertile place inclined towards the south.
2. All hair, including head and facial hair, was removed from the body.
3. A funeral procession of four parts was organized. The immediate family members carried the sacred fire and the sacrificial vessels. Behind them an odd number of persons carried the dead body. Next, a cow or goat, preferably black in color, followed. Finally, the relatives and friends of the dead person followed.
4. Once the funeral pyre had been prepared the body was placed on sacred grass that lined the inside of the cremation pit along with wood. In the case of a husband who had died, his wife would lie to the north side of his body. A brother or some other representative of the deceased would ask her to rise before the fire was lighted. The sacrificial implements used by the deceased person would also be placed alongside the body.
5. The body would be covered with the skin of an anustarani cow or goat. If there was no animal then cakes of  rice would serve the purpose.
6. The fire was lighted starting at the head.
7. When the entire body had been consumed, the mourners would circumambulate in a counter-clockwise direction and then leave without looking back. They would then go and bathe.

During the Puranic period the procedures were as follows.

1. At the time of death sacred verses were recited to revive the dead person. When these had failed the priest would announce the death. The cremation, if possible, was to be performed on the day of the death.
2. Professional mourners would be hired, who would gather around the deceased with disheveled hair, disordered garments, and dust covered bodies and begin wailing and sobbing.
3. The body was washed; the hair and nails were cut. The body was dressed in new garments and adorned with ornaments.
4. The body was carried on the shoulders of relatives, or pulled in a cart, followed by mourners who would recite sacred prayers until they reached the cremation site.
5. After arriving at the cremation site the body would be placed on the funeral pyre with the head facing the south.
6. The chief mourner placed ghee on the body to the accompaniment of sacred prayers.
7. All jewels and ornaments were removed from the body and a small mound of cow dung was placed on the stomach or chest. The chief mourner walked around the body three times in a counterclockwise direction while sprinkling water from an earthen vessel. The vessel was then broken on the ground near the head of the deceased.
8. The chief mourner lighted the fire at the head to the accompaniment of prayers.
9. Prayers were recited to direct the various parts of the deceased's body to merge with the universal elements: the voice to the sky, the eyes to the sun, the vital breath to the wind, and so forth.
10. After the fire has consumed the body the mourning party returned home to bathe and purify themselves with prayers for peace.
11. Three days after the cremation the chief mourner returned to the burning area and ceremonially sprinkled the ashes with water. The ashes were later poured into the Ganges or other sacred body of water in a ceremony called visarjanam.

Current cremation practice in India generally follows this Puranic model.

Burning in Effigy (kusa-puttalika-daha)

If a person had died but the body could not be reclaimed, as in the case of a person who had drowned or had been killed in battle, it was still absolutely essential for a cremation to take place. The reason was simple: without cremation the departed soul could not begin the transition into a pitr. In lieu of a body an image could be cremated. The Bhavisya-purana describes an image made of 360 strands of kusa, a kind of sacred grass: Forty for the head, twenty for the neck, one hundred in the two arms, twenty in the chest, twenty in the belly, thirty in the hips, one hundred in the two thighs, and thirty in the knees and shanks. Another account uses a coconut for the head, a bottle gourd for the mouth, five gems for the teeth, a plantain for the tongue, two shells for the eyes, clay for the nose, plantain leaves for the ears, the shoots of the fig tree for the hair, lotus fibers for the entrails, earth and barley paste for the flesh, honey for the blood, the skin of an antelope for the skin, a lotus for the naval, eggplant for the scrotum, and tree bark for garments!

If a person became missing, but was not specifically known to be dead, as in the case of someone who had gone to a foreign land and not returned, the relatives were advised to wait 12 years before performing the cremation. In the case of a person who has been cremated in effigy, but who then returned home, the person needed to be reborn by being passed through the legs of a female and then, step-by-step, have all the purificatory ceremonies (samskaras) performed. This may even including a re-marriage if necessary.

There was also a special rite called Narayana-bali that was performed when a person had died under unusual circumstances, such as through suicide or accidental death. The Narayana-bali was atonement for the situation and made the deceased fit for receiving the regular funeral process and subsequent rites.

The Anustarani Animal

Both the Rg and the Atharva Vedas prescribe that the skin and organs of a cow or she-goat, called an anustarani animal, be burned along with the body. This was done in order to lessen the pain inflicted on the departing soul by the scorching fire. The hide of the animal covered the body. The vital organs of the animal were placed in the hands and around the body of the deceased. During the Grhya period this practice declined and by the Puranic period was stopped altogether. Instead, rice was spread around the body in lieu of the skin. During Ravana's funeral Valmiki describes how an anustarani animal was used.

There is an interesting story in the Aitreya-brahmana that tells how rice became the substitute for the anustarani animal. "In the beginning the gods used human beings for sacrifice. Overtime the sap of life left the human being and entered the body of the horse. Thereafter, the horse became the object of sacrifice. In time this sap of life left the horse and entered the ox. The ox became the object of sacrifice. Then again when the sap of life left the ox and entered sheep, a sheep became the object of sacrifice. Soon this sap of life left the sheep and entered the goat, wherein the goat became the object of sacrifice. For a long time the goat remained the object of sacrifice. Eventually, the sap of life left the goat and entered the earth. Thereupon, the earth became rice and rice became the fit substitute for the sap of life." Here we get the history of the sacrificial animal and the relationship between rice and the sacrificial animal.

Post cremation Burial (Pitr-medha)

 During the Vedic and early Grhya periods it was common to bury the incinerated bones of a deceased person in an urn. This was the pitr-medha ceremony. The Grhya-sutras of Asvalayana describe how the burned bones were to be collected on the third lunar day (tithi) after death. In the case of a man who had died, the bones were to be collected by elderly men and placed into a male urn. In the case of a woman, the bones were to be collected by elderly women and placed into a female urn. Urns were designed by their shape to be male or female. The performers of this ceremony were to walk three times in a counterclockwise direction around the bones while sprinkling milk and water from a particular kind of twig (sami). The bones were then placed into the urn as they were picked up individually with the thumb and fourth finger. First the bones of the feet were to be gathered and then successively the other bones were to be gathered working toward the head. After the bones had been purified and gathered they were sealed and buried in a secure location.

By the end of the Grhya period the practice of burying bones in an urn declined.