Dashavatara (/ˌdæʃəˈvætərə/; Sanskrit: दशावतार, daśāvatāra) refers to the ten primary avatars of Vishnu, the Hindu god of preservation. Vishnu is said to descend in form of an avatar to restore cosmic order. The word Dashavatara derives from daśa, meaning 'ten', and avatar (avatāra), roughly equivalent to 'incarnation'.
The word Dashavatara derives from daśa, meaning 'ten' and avatar (avatāra), meaning 'incarnation'.
Various versions of the list of Vishnu's avatars exist, varying per region and tradition. Some lists mention Krishna as the eighth avatar and the Buddha as the ninth avatar, while others – such as the Yatindramatadipika, a 17th-century summary of Srivaisnava doctrine – give Balarama as the eighth avatar and Krishna as the ninth. The latter version is followed by some Vaishnavas who don't accept the Buddha as an incarnation of Vishnu. Though no list can be uncontroversially presented as standard, the "most accepted list found in Puranas and other texts is [...] Krishna, Buddha."[note 1]
The following table summarises the position of avatars within the Dashavatara in many but not all traditions:
1st to 5th of the Dashavatars on Udupi temple gopuram, Karnataka.
1 - Matsya, the fish. Vishnu takes the form of a fish to save Manu from the deluge (Pralaya), after which he takes his boat to the new world along with one of every species of plant and animal, gathered in a massive cyclone.
2 - Kurma, the giant tortoise. When the devas and asuras were churning the Ocean of milk in order to get Amrita, the nectar of immortality, the mount Mandara they were using as the churning staff started to sink and Vishnu took the form of a tortoise to bear the weight of the mountain.
3 - Varaha, the boar. He appeared to defeat Hiranyaksha, a demon who had taken the Earth, or Prithvi, and carried it to the bottom of what is described as the cosmic ocean (much like in ether theory) in the story. The battle between Varaha and Hiranyaksha is believed to have lasted for a thousand years, which the former finally won. Varaha carried the Earth out of the ocean between his tusks and restored it to its place in the universe.
4 - Narasimha, the half-man/half-lion. The rakshasa (Demon) Hiranyakashyapa, the elder brother of Hiranyaksha, was granted a powerful boon from Brahma that he could not be killed by man or animal, inside or outside a room, during day or night, neither on ground nor in air, with a weapon that is either living or inanimate. Hiranyakashipu persecuted everyone for their religious beliefs including his son who was a Vishnu follower. Vishnu descended as an anthropomorphic incarnation, with the body of a man and head and claws of a lion. He disemboweled Hiranyakashipu at the courtyard threshold of his house, at dusk, with his claws, while he lay on his thighs. Narasimha thus destroyed the evil demon and brought an end to the persecution of human beings including his devotee Prahlada, according to the Hindu mythology.
5 - Vamana, the dwarf. The fourth descendant of Vishnu, Bali, with devotion and penance was able to defeat Indra, the god of firmament. This humbled the other deities and extended his authority over the three worlds. The gods appealed to Vishnu for protection and he descended as a boy Vamana. During a yajna (यज्ञ) of the king, Vamana approached him and Bali promised him for whatever he asked. Vamana asked for three paces of land. Bali agreed, and the dwarf then changed his size to that of a giant Trivikrama form. With his first stride he covered the earthly realm, with the second he covered the heavenly realm thereby symbolically covering the abode of all living beings. He then took the third stride for the netherworld. Bali realized that Vamana was Vishnu incarnate. In deference, the king offered his head as the third place for Vamana to place his foot. The avatar did so and thus granted Bali immortality and making him ruler of Pathala, the netherworld. This legend appears in hymn 1.154 of the Rigveda and other Vedic as well as Puranic texts.
6 - Parashurama, the warrior with the axe. He is son of Jamadagni and Renuka and was granted as boon, an axe after a penance to Shiva. He is the first Brahmin-Kshatriya in Hinduism, or warrior-sage, who had to follow the Dharma of both, a Brahmin as well as a Kshatriya. Once, when king Kartavirya Arjuna and his hunting party halted at the ashrama of Jamadagni, the father of Parashurama, and the sage was able to feed them all with the aid of the divine cow Kamadhenu. The king demanded the cow, but Jamadagni refused. Enraged, the king took it by force and destroyed the ashram. Parashurama then killed the king at his palace and destroyed his army. In revenge, the sons of Kartavirya killed Jamadagni. Parashurama took a vow to kill every Kshatriya on earth twenty-one times over, and filled five lakes with their blood. Ultimately, his grandfather, rishi Rucheeka, appeared before him and made him halt. He is a chiranjeevi (immortal), and believed to be alive today in penance at Mahendragiri. He also credited for creating Kerala by throwing his mighty axe as per Hindu mythology. The place the axe landed in sea got its water displaced and the land which emerged thus came to be known as Kerala.
7 - Rama, the prince and king of Ayodhya. He is a commonly worshipped avatar in Hinduism, and is thought of as the ideal model of a common prince without super powers, despite being an incarnation. His story is recounted in one of the most widely read scriptures of Hinduism, the Ramayana. While in exile from his own kingdom with his brother Lakshman and the God Hanuman, his wife Sita was abducted by the demon king of Lanka, Ravana. He travelled to Lanka, killed the demon king and saved Sita. Rama and Sita returned home and were crowned. But later, due to constant tantrums by his citizens, he exiled Sita and his unborn sons out of his kingdom. Rama had won one of the greatest and most honest wars, but still lived most of his adult life in exile or in misery.
 - Balarama, the elder brother of Krishna, is regarded generally as an avatar of Shesha an extension of Ananta, a form of Lord Vishnu . Balarama is included as the eighth avatar of Vishnu in the Sri Vaishnava lists, where Buddha is omitted and Krishna appears as the ninth avatar in this list. He is particularly included in the lists where Krishna is removed and becomes the source of all.
[9 or 8 or "0"] - Krishna was the eighth son of Devaki and Vasudeva. A frequently worshipped deity in Hinduism, he is the hero of various legends and embodies qualities such as love and playfulness. In some lists Krishna is removed and becomes the source of all. Krishna is the mirror opposite of Rama. Rama was a strict follower of honesty, integrity and monogamy, while Krishna had 8 wives, and used any available means to get to his goal.
 - Buddha: Gautama Buddha, the founder of Buddhism, is commonly included as an avatar of Vishnu in Hinduism. Buddha is sometimes depicted in Hindu scriptures as a preacher who deludes and leads demons and heretics away from the path of the Vedic scriptures, but another view praises him a compassionate teacher who preached the path of ahimsa (non-violence).[note 1]
 - In certain Odia literary creations from Odisha, Jagannath has been treated as the Ninth avatar, by substituting Buddha.
10 - Kalki is described as the final incarnation of Vishnu, who appears at the end of each Kali Yuga. He will be atop a white horse and his sword will be drawn, blazing like a comet. He appears when only chaos, evil and persecution prevails, dharma has vanished, and he ends the kali yuga to restart Satya Yuga and another cycle of existence.
The Buddha was included as one of the avatars of Vishnu under Bhagavatism by the Gupta period between 330 and 550 CE. The mythologies of the Buddha in the Theravada tradition and of Vishnu in Hinduism share a number of structural and substantial similarities. For example, states Indologist John Holt, the Theravada cosmogony and cosmology states the Buddha covered 6,800,000 yojanas in three strides, including earth to heaven and then placed his right foot over Yugandhara – a legend that parallels that of the Vamana avatar in Hinduism. Similarly, the Buddha is claimed in the Theravada mythology to have been born when dharma is in decline, so as to preserve and uphold the dharma. These similarities may have contributed to the assimilation of the Buddha as an avatar of Vishnu.
The adoption of Buddha as an avatar in Bhagavatism was a catalyzing factor in Buddhism's assimilation into Vaishnavism's mythic hierarchy. By the 8th century CE, the Buddha was included as an avatar of Vishnu in several Puranas. This assimilation is indicative of the Hindu ambivalence toward the Buddha and Buddhism.[note 10] Conversely, Vishnu has also been assimilated into Sinhalese Buddhist culture, and Mahayana Buddhism is sometimes called Buddha-Bhagavatism. By this period, the concept of Dashavatara was fully developed.
Temple door depicting Dashavatar with Vithoba, at Sree Balaji Temple, Goa. From leftmost upper corner, clock wise: Matsya, Narasimha, Parashurama, Rama, Krishna, Kalki, Vamana, Vithoba, Varaha and Kurma.
In traditions that emphasize the Bhagavata Purana, Krishna is the original Supreme Personality of Godhead, from whom everything else emanates. Gaudiya Vaishnavas worship Krishna as Svayam Bhagavan, or source of the incarnations. The Vallabha Sampradaya and Nimbarka Sampradaya, (philosophical schools) go even further, worshiping Krishna not only as the source of other incarnations, but also Vishnu himself, related to descriptions in the Bhagavata Purana. Mahanubhavas also known as the Jai Kishani Panth, considers Lord Krishna as the supreme God and don't consider the list of Dashavatara while consider another list of Panchavatara (5 Avatars). 
Thirty-nine avatars are mentioned in the Pañcaratra including the likes of Garuda.  However, despite these lists, the commonly accepted number of ten avatars for Vishnu was fixed well before the 10th century CE.
Some modern interpreters interpret Vishnu's ten main avatars as an ascending order from simple life-forms to more complex life-forms, and see the Dashavataras as a reflection, or a foreshadowing, of the modern theory of evolution. Such an interpretation was first propounded by TheosophistHelena Blavatsky in her 1877 opus Isis Unveiled, in which she proposed the following ordering of the Dashavataras:
The Puranas speak of the different manifestations or incarnations of the Deity in different epochs of the world history. Lo! The Hindu Avatar rises from the lowest scale of life through the fish, the tortoise, and the hog up to the perfection of humanity. Indian Avatarism is, indeed, a crude representation of the ascending scale of Divine creation. Such precisely is the modern theory of evolution.
Similarly Aurobindo regarded "Avataric Evolutionism" as a "parable of evolution", one which does not endorse evolutionism, but hints at "transformative phases of spiritual progress". According to Nanda, the Dashavatara concept has led to some Hindus asserting that their religion is more open to scientific theories, and has not opposed or persecuted scientists midst them like the way Christianity and Islam has. But, adds Nanda, Hinduism has many cosmological theories and even the Vaishnava one with Dashavatara concept does not explicitly teach evolution of species, rather it states an endless cycles of creationism.
Dasavatar, 19th century
The Dashavatara concept appealed to other scholars. Monier Monier-Williams wrote "Indeed, the Hindus were ... Darwinians centuries before the birth of Darwin, and evolutionists centuries before the doctrine of evolution had been accepted by the Huxleys of our time, and before any word like evolution existed in any language of the world."J. B. S. Haldane suggested that Dashavatara gave a "rough idea" of vertebrate evolution: a fish, a tortoise, a boar, a man-lion, a dwarf and then four men (Kalki is not yet born).Nabinchandra Sen explains the Dashavatara with Darwin's evolution in his Raivatak.C. D. Deshmukh also remarked on the "striking" similarity between Darwin's theory and the Dashavatara.
Some Vaishnava Hindus reject this "Avataric Evolutionism" concept. For example, ISKCON's Prakashanand states that this apologeticism degrades the divine status of Rama and Krishna, unduly sequences Rama as inferior to Krishna, both to the Buddha. Rama and Krishna are supremely divine, each right and perfect for the circumstances they appeared in, states Prakashanand.
Doniger O'Flaherty, Wendy (1994), Hindu Myths, Penguin Books India, ISBN9780144000111, p.175. Doniger: "Visnu is generally said to have had ten incarnations [...] Krsna [...] the Buddha."
Flood, Gavin D. (1996), An Introduction to Hinduism, Cambridge University Press, ISBN978-0-521-43878-0, p.116. Flood: "...by the eight century the standard number of descent-forms in the Vaisnava Puranas is ten. These are [...] Krsna, Buddha."
Klostermaier, Klaus K. (2007), Hinduism: A Beginner's Guide, Oneworld Publications, "Visnu Avataras". Klostermaier: "The most common tradition speaks of ten such avataras [...] Krsna [...] Buddha."
Krishna, Nanditha (2010), The Book of Vishnu, Penguin Books India, ISBN9780143067627, p.28-29. Krishna: "Krishna [...] Buddha [...] There is a difference of opinion as to whether Buddha was an incarnation of Vishnu [...] The alternative then is Balarama, Rama of the plough and elder brother of Krishna, who is listed after Rama, thereby removing Buddha and making Krishna the ninth incarnation."
Holt, John Clifford (2008), The Buddhist Viṣṇu: Religious Transformation, Politics, and Culture, p.14-15; p.372 note 9 refers to four Purana's which mention the Buddha in 9th position: Varaha Purana 4.2; Matsya Purana 285.6-7; Agni Purana 49.8; Bhagavata Purana X.40.22 and I.3.
^prabhat Mukherjee: Orissa; Leyden: West Bengal Orissa.
^Donald J. LaRocca, Metropolitan Museum of Art, describes a katar with Rama-Krishna-Buddha, referring to Rama as Ramachandra, or alternately Balarama. Yet, Hoiberg specifically states that Rama, as an avatar of Vishnu, is Ramachandra.
^This 7th century (or early 8th century) inscription is significant for several reasons. It is the earliest known stone inscription about the ten avatars of Vishnu, and prior to that they are found in older texts. The stone inscription mentions the Buddha as an avatar of Vishnu in a Hindu temple. It also does not mention Krishna, but Balarama consistent with old Hindu and Jain texts of South India, the former equating Krishna to be identical to Vishnu.
^Some Hindus believe that sugata is not an ephitet for Gautama Buddha, and that "Sugata Buddha" refers to the Buddha-avatar of Vishnu. In this view, Sugata Buddha and Gautama Buddha were two different persons. This way, "There is no evidenc whatsoever that Sugata Buddha, Lord Vishnu's incarnation, was in any way connected with atheism."
^Flood, Gavin D. (1996). An introduction to Hinduism. Cambridge, UK: Cambridge University Press. p. 341. ISBN978-0-521-43878-0. Retrieved 21 April 2008."Early Vaishnava worship focuses on three deities who become fused together, namely Vasudeva-Krishna, Krishna-Gopala and Narayana, who in turn all become identified with Vishnu. Put simply, Vasudeva-Krishna and Krishna-Gopala were worshiped by groups generally referred to as Bhagavatas, while Narayana was worshipped by the Pancaratra sect."
1 The list of ten avatars varies regionally. The two substitutions involve Balarama, Krishna and Buddha is considered the avatar of Vishnu. Krishna is almost always included; in exceptions, he is considered the source of all avatars.