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Apart from the well known Dashavatra, Mahavishnu has taken many incarnations. Two readily come to the mind as Hari for liberation of Gajendra and Hu Hu and as Kapila Muni giving knowledge to His mother Devahuti.

How many Avataaras are enumerated and described in Srimad Bhagavatam, Harivamsam and other Puranaas? What is the number of His Avataaras?

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I am writing the answer based on Shrimada Bhagavatam only. In Shrimada Bhagavatam, Canto 11, Chapter 4, Sage Drumila had described various incarnations of Lord Vishnu to King Nimi. But these are not the exact incarnations as said by Sage Drumila that the incarnation of Lord are infinite.

SB 11.4.3 — When the primeval Lord Nārāyaṇa created His universal body out of the five elements produced from Himself and then entered within that universal body by His own plenary portion, He thus became known as the Puruṣa.

SB 11.4.4 — Within His body are elaborately arranged the three planetary systems of this universe. His transcendental senses generate the knowledge-acquiring and active senses of all embodied beings. His consciousness generates conditioned knowledge, and His powerful breathing produces the bodily strength, sensory power and conditioned activities of the embodied souls. He is the prime mover, through the agency of the material modes of goodness, passion and ignorance. And thus the universe is created, maintained and annihilated.

SB 11.4.5 — In the beginning, the original Supreme Personality manifested the form of Brahmā through the material mode of passion in order to create this universe. The Lord manifested His form as Viṣṇu, the Lord of sacrifice and protector of the twice-born brāhmaṇas and their religious duties, to maintain the universe. And when the universe is to be annihilated the same Supreme Lord employs the material mode of ignorance and manifests the form of Rudra. The created living beings are thus always subject to the forces of creation, maintenance and destruction.

SB 11.4.6 — Nara-Nārāyaṇa Ṛṣi, who is perfectly peaceful and is the best of sages, was born as the son of Dharma and his wife Mūrti, the daughter of Dakṣa. Nara-Nārāyaṇa Ṛṣi taught the devotional service of the Lord, by which material work ceases, and He Himself perfectly practiced this knowledge. He is living even today, His lotus feet served by the greatest of saintly persons.

SB 11.4.17 — The infallible Supreme Personality of Godhead, Viṣṇu, has descended into this world by His various partial incarnations such as Lord Haṁsa [the swan], Dattātreya, the four Kumāras and our own father, the mighty Ṛṣabhadeva. By such incarnations, the Lord teaches the science of self-realization for the benefit of the whole universe. In His appearance as Hayagrīva He killed the demon Madhu and thus brought the Vedas back from the hellish planet Pātālaloka.

SB 11.4.18 — In His appearance as a fish, the Lord protected Satyavrata Manu, the earth and her valuable herbs. He protected them from the waters of annihilation. As a boar, the Lord killed Hiraṇyākṣa, the son of Diti, while delivering the earth from the universal waters. And as a tortoise, He lifted Mandara Mountain on His back so that nectar could be churned from the ocean. The Lord saved the surrendered king of the elephants, Gajendra, who was suffering terrible distress from the grips of a crocodile.

SB 11.4.19 — The Lord also delivered the tiny ascetic sages called the Vālakhilyas when they fell into the water in a cow’s hoofprint and Indra was laughing at them. The Lord then saved Indra when Indra was covered by darkness due to the sinful reaction for killing Vṛtrāsura. When the wives of the demigods were trapped in the palace of the demons without any shelter, the Lord saved them. In His incarnation as Nṛsiṁha, the Lord killed Hiraṇyakaśipu, the king of demons, to free the saintly devotees from fear.

SB 11.4.20 — The Supreme Lord regularly takes advantage of the wars between the demons and demigods to kill the leaders of the demons. The Lord thus encourages the demigods by protecting the universe through His various incarnations during the reigns of each Manu. The Lord also appeared as Vāmana and took the earth away from Bali Mahārāja on the plea of begging three steps of land. The Lord then returned the entire world to the sons of Aditi.

SB 11.4.21 — Lord Paraśurāma appeared in the family of Bhṛgu as a fire that burned to ashes the dynasty of Haihaya. Thus Lord Paraśurāma rid the earth of all kṣatriyas twenty-one times. The same Lord appeared as Rāmacandra, the husband of Sītādevī, and thus He killed the ten-headed Rāvaṇa, along with all the soldiers of Laṅkā. May that Śrī Rāma, whose glories destroy the contamination of the world, be always victorious.

SB 11.4.22 — To diminish the burden of the earth, the unborn Lord will take birth in the Yadu dynasty and perform feats impossible even for the demigods. Propounding speculative philosophy, the Lord, as Buddha, will bewilder the unworthy performers of Vedic sacrifices. And as Kalki the Lord will kill all the low-class men posing as rulers at the end of the Age of Kali.

SB 11.4.23 — O mighty-armed King, there are innumerable appearances and activities of the Supreme Lord of the universe similar to those I have already mentioned. In fact, the glories of the Supreme Lord are unlimited.

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Lord's incarnations are infinite. Among all the Avataaras, 10 (or Dashavatara) and 24 (Chaubis Avtar) Avataaras are more famous. The list of these famous 10 and 24 incarnation varies from one text to another.

Here is the list of Lord Vishnu's few incarnations:

  • Varaha

  • Kumāras (from Bhāgavata Purāṇ)

  • Nārada (from Bhāgavata Purāṇ)

  • Nārāyaṇa (from Bhāgavata Purāṇ)

  • Yajna (from Vishnu Purana)

  • Ajita (from Vishnu Purana)

  • Satya (from Vishnu Purana)

  • Hari (from Vishnu Purana)

  • Mánasa (from Vishnu Purana)

  • Vaikuntha (from Vishnu Purana)

  • Kurma

  • Dhanvantari (from Bhāgavata Purāṇ)

  • Mohini

  • Hayagreeva (from Bhāgavata Purāṇ)

  • Matsya

  • Nrisimha

  • Vámana

  • Kapila (from Bhāgavata Purāṇ)

  • Dattatreya (from Bhāgavata Purāṇ)

  • Haṁsa (from Bhāgavata Purāṇ)

  • Ṛṣabha (from Bhāgavata Purāṇ)

  • Pṛthu (from Bhāgavata Purāṇ)

  • Mandhata (from Vayu Purana)

  • Parashurama

  • Rama

  • Krishna Dvaipāyana (28th Vyasa) (from Bhāgavata Purāṇ)

  • Balarāma (from Bhāgavata Purāṇ)

  • Krishna

  • Buddha (from Bhāgavata Purāṇ)

  • Kalki

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