The number of such incarnations is considered to be infinite (ananta). However, the Sattvata Samhita which is the oldest Pancharatra text, enumerates thirty-nine as important avataras. This is also stated by the Ahirbudhanya Samhita. The Visvaksena Samhita mentions thirty-six avataras. As there is some difference of opinion among the Sarhhitas regarding the exact number, Vedanta Desika states that the total number of vibhavas is above 30 and below 40.
These avataras are classified under two categories: principal or important ones (mukhya) and secondary or subsidiary (gaund). The difference is based on the fact that the former refers to the direct incarnations (saksat avatara), that is, Vishnu Himself with a transcendental body assuming the human form and the latter represents the entry of the divine power into the bodies of the specific individuals. The latter is also called Dvesa avatara. Avesa means to get possessed and when an individual becomes possessed of the divine spirit it is taken as dvesavatdra. This is of two kinds: (1) the very spiritual divine body itself can pervade the body of another individual as in the case of Parasurama (one of the ten Vishnu incarnations), in whom God pervaded his body; (2) the divine power (shakti) is infused into the body of an individual to make the latter carry out the extraordinary acts of creation or dissolution of the universe, as in the case of Brahma and Rudra. Sage Vyasa, the author of the Mahabharata, is considered to be a case of amsavatara.
According to the Vaishnava tradition, the ten Alvars are anupravesavataras of God. In the same way, the important Vaishnava acharyas such as Ramanuja, Vedanta Desika and others are regarded as the incarnations of either the divine elements or the very Godhead. The basis for this belief is the statement in the Jayakhya Sarhhita, one of the oldest Pancharatra treatises. It says that Narayana, the Supreme Being assumes the bodies of human beings in order to uplift the human beings immersed in the ocean of bondage through the aid of shastra, out of His compassion. All the incarnations of God, both direct ones as well as the indirect ones, take place out of His own will (iccha) for the purpose of protection of devotees and destruction of evil, as already explained. The bodies assumed during incarnation are of spiritual character and, therefore, they are not defiled by any kind of defects.
The thirty-nine avataras, as enumerated in the Ahirbhudhnya Samhitai are the following: Padmanabha, Dhruva, Ananta, Saktyatman, Madhusudana, Vidyadhideva, Kapila, Visvarupa, Vihangama, Krodatman, Badabavakira, Dharma, Vagisvara, Ekarrtavasayin, Kamatesvara, Varaha, Narasimha, Piyusahararia, Sripati, Kantatman, Rahujit, Kalanemighna, Parijatahara, Lokanatha, Santatman, Dattatreya, Nyagrodhasayin, Ekasmgatanu, Vamanadeha, Nara, Narayana, Hari, Trivikrama, Krishna, Parasurama, Rama, Vedavid, Kalkin and Patalasayana.
The origin and nature of each one of these vibhavas is shrouded with mysterious mythological episodes. Certain observations may be made regarding the features of these vibhavas. The long list of avataras has taken place in the remote past (time is infinite according to Hindu Philosophy) at different epochs (kalpas). Some of them such as Padmanabha, or the form of Vishnu from whose naval the lotus grew and from which Brahma was born, goes back to the beginning of the creation of the universe. The avataras of Vishnu as Ekarnavasayin or as sleeping on the primeval waters and Nyagrodhasayin or Vishnu as a boy floating on nyagrodha leaf in whose mouth the Sage Markandeya discovered the dissolved universe belong to the period when there was the great deluge after the entire universe was dissolved (pralaya). The Vedic Mythology refers to the Matsya Avatara or Vishnu's incarnation as a fish, Kurma avatara or incarnation as a tortoise and Varaha avatara or the descent of Vishnu as a boar to uplift the earth submerged in water.
From the details of the Vedic passages these avataras must have taken place either prior to creation or after the dissolution of the universe. Going by the details furnished by the Puranas, the avataras occurred at different yugas. The avataras of Vamana or Visnu as a dwarf to punish the powerful demon-king Bali, and Narasimha, Vishnu in the form of man-lion to kill the demon Hiranyakasipu and protect the ardent devotee, Prahlada, took place in the Kritayuga. The incarnation of Lord Rama or Vishnu born as the son of the emperor, Dasaratha to kill the demon Havana, occurred in Tretayuga. The avatara of Krishna happened in Dvaparayuga. Apart from the time factor, what is important to note is that the avataras are not confined to human forms as in the case of Vamana, Rama, Krishna and Parasurama. They cover the animal forms too, such as fish, tortoise, boar, swan (hamsa) and a combination of man and animal, as in the case of Hayagriva (man with the head of the horse) and Narasimha (man-lion). Even the plant kingdom is chosen for an avatara, as in the case of a crooked mango-tree in the Dandaka forest mentioned in the Visvaksena Samhita. This establishes the fact that all livings beings in the universe which are God's vibhutis or glory are sacred. Though God chooses the human beings and animals as objects for His incarnation, He does not become defiled by His association with them, because, as already explained, He enters into them with His spiritual divine body (divyamangala-vigraha) out of His will (iccha) and remains unaffected by their defects.