The Lord incarnated Himself as Dattātreya, the son of Ṛṣi Atri and Anasūyā. The history of the birth of Dattātreya as an incarnation of the Lord is mentioned in the Brahmāṇḍa Purāṇa in connection with the story of the devoted wife. It is said there that Anasūyā, the wife of Ṛṣi Atri, prayed before the Lords Brahmā, Viṣṇu and Śiva as follows: "My lords, if you are pleased with me, and if you desire me to ask from you some sort of blessings, then I pray that you combine together to become my son."
(Srimad Bhagavatam------1:3:11------purport).

Atri: Atri Muni was a great brāhmaṇa sage and was one of the mental sons of Brahmājī. Brahmājī is so powerful that simply by thinking of a son he can have it. These sons are known as mānasa-putras. Out of seven mānasa-putras of Brahmājī and out of the seven great brāhmaṇa sages, Atri was one. In his family the great Pracetās were also born. Atri Muni had two kṣatriya sons who became kings. King Arthama is one of them. He is counted as one of the twenty-one prajāpatis. His wife's name was Anasūyā, and he helped Mahārāja Parīkṣit in his great sacrifices.
(Srimad Bhagavatam------1:9:6:7------translation).

The great sage Atri prayed for offspring, and the Lord, being satisfied with him, promised to incarnate as Atri's son, Dattātreya (Datta, the son of Atri). And by the grace of the lotus feet of the Lord, many Yadus, Haihayas, etc., became so purified that they obtained both material and spiritual blessings.
Transcendental relations between the Personality of Godhead and the living entities are eternally established in five different affectionate humors, which are known as śānta, dāsya, sakhya, vātsalya and mādhurya. The sage Atri was related with the Lord in the affectionate vātsalya humor, and therefore, as a result of his devotional perfection, he was inclined to have the Personality of Godhead as his son. The Lord accepted his prayer, and He gave Himself as the son of Atri. Such a relation of sonhood between the Lord and His pure devotees can be cited in many instances. And because the Lord is unlimited, He has an unlimited number of father-devotees. Factually the Lord is the father of all living entities, but out of transcendental affection and love between the Lord and His devotees, the Lord takes more pleasure in becoming the son of a devotee than in becoming one's father. The father actually serves the son, whereas the son only demands all sorts of services from the father; therefore a pure devotee who is always inclined to serve the Lord wants Him as the son, and not as the father. The Lord also accepts such service from the devotee, and thus the devotee becomes more than the Lord. The impersonalists desire to become one with the Supreme, but the devotee becomes more than the Lord, surpassing the desire of the greatest monist. Parents and other relatives of the Lord achieve all mystic opulences automatically because of their intimate relationship with the Lord. Such opulences include all details of material enjoyment, salvation and mystic powers. Therefore, the devotee of the Lord does not seek them separately, wasting his valuable time in life. The valuable time of one's life must therefore be fully engaged in the transcendental loving service of the Lord. Then other desirable achievements are automatically gained. But even after obtaining such achievements, one should be on guard against the pitfall of offenses at the feet of the devotees. The vivid example is Haihaya, who achieved all such perfection in devotional service but, because of his offense at the feet of a devotee, was killed by Lord Paraśurāma. The Lord became the son of the great sage Atri and became known as Dattātreya.
(Srimad Bhagavatam------2:7:4------translation and purport).

The nine principal ṛṣis, or sages, are Marīci, Atri, Aṅgirā, Pulastya, Pulaha, Kratu, Bhṛgu, Vasiṣṭha and Atharvā. All these ṛṣis are most important, and Brahmā desired that the nine daughters already born of Kardama Muni be handed over to them. Here two words are used very significantly-yathā-śīlam and yathā-ruci. The daughters should be handed over to the respective ṛṣis, not blindly, but according to the combination of character and taste. That is the art of combining a man and woman. Man and woman should not be united simply on the consideration of sex life. There are many other considerations, especially character and taste. If the taste and character differ between the man and woman, their combination will be unhappy. Even about forty years ago, in Indian marriages, the taste and character of the boy and girl were first of all matched, and then they were allowed to marry. This was done under the direction of the respective parents. The parents used to astrologically determine the character and tastes of the boy and girl, and when they corresponded, the match was selected: "This girl and this boy are just suitable, and they should be married." Other considerations were less important. The same system was also advised in the beginning of the creation by Brahmā: "Your daughters should be handed over to the ṛṣis according to taste and character."
(Srimad Bhagavatam------3:24:15------purport).

It appears that the great sage Atri Muni had no specific idea of the Supreme Personality of Godhead. Of course, he must have been conversant with the Vedic information that there is a Supreme Personality of Godhead who is the creator of the universe, from whom everything emanated, who maintains this created manifestation, and in whom the entire manifestation is conserved after dissolution. Yato vā imāni bhūtāni (Taittirīya Upaniṣad 3.1.1). The Vedic mantras give us information of the Supreme Personality of Godhead, so Atri Muni concentrated his mind upon that Supreme Personality of Godhead, even without knowing His name, just to beg from Him a child exactly on His level. This kind of devotional service, in which knowledge of God's name is lacking, is also described in Bhagavad-gītā where the Lord says that four kinds of men with backgrounds of pious activities come to Him asking for what they need. Atri Muni wanted a son exactly like the Lord, and therefore he is not supposed to have been a pure devotee, because he had a desire to be fulfilled, and that desire was material. Although he wanted a son exactly like the Supreme Personality of Godhead, this desire was material because he did not want the Personality of Godhead Himself, but only a child exactly like Him. If he had desired the Supreme Personality of Godhead as his child, he would have been completely free of material desires because he would have wanted the Supreme Absolute Truth, but because he wanted a similar child, his desire was material. Thus Atri Muni cannot be counted among the pure devotees.
(Srimad Bhagavatam------4:1:20------purport).

Daṇḍa means "a long rod," and vat means "like." Before a superior, one has to fall down on the ground just like a stick, and this sort of offering of respect is called daṇḍavat. Atri Ṛṣi offered his respect to the three deities in that way. They were identified by their different carriers and different symbolic representations. In that connection it is stated here that Lord Viṣṇu was sitting on Garuḍa, a big aquiline bird, and was carrying in His hand a disc, Brahmā was sitting on a swan and had in his hand kuśa grass, and Lord Śiva was sitting on a bull and carrying in his hand a small drum called a ḍamaru. Atri Ṛṣi recognized them by their symbolic representations and different carriers, and thus he offered them prayers and respects.
(Srimad Bhagavatam------4:1:24------purport).

But since his heart was already attracted by the deities, somehow or other he gathered his senses, and with folded hands and sweet words he began to offer prayers to the predominating deities of the universe. The great sage Atri said: O Lord Brahmā, Lord Viṣṇu and Lord Śiva, you have divided yourself into three bodies by accepting the three modes of material nature, as you do in every millennium for the creation, maintenance and dissolution of the cosmic manifestation. I offer my respectful obeisances unto all of you and beg to inquire whom of you three I have called by my prayer.
Atri Ṛṣi called for the Supreme Personality of Godhead, jagad-īśvara, the Lord of the universe. The Lord must exist before the creation, otherwise how could He be its Lord? If someone constructs a big building, this indicates that he must have existed before the building was constructed. Therefore the Supreme Lord, the creator of the universe, must be transcendental to the material modes of nature. But it is known that Viṣṇu takes charge of the mode of goodness, Brahmā takes charge of the mode of passion, and Lord Śiva takes charge of the mode of ignorance. Therefore Atri Muni said, "That jagad-īśvara, the Lord of the universe, must be one of you, but since three of you have appeared, I cannot recognize whom I have called. You are all so kind. Please let me know who is actually jagad-īśvara, the Lord of the universe." In fact, Atri Ṛṣi was doubtful about the constitutional position of the Supreme Lord, Viṣṇu, but he was quite certain that the Lord of the universe cannot be one of the creatures created by māyā. His very inquiry about whom he had called indicates that he was in doubt about the constitutional position of the Lord. Therefore he prayed to all three, "Kindly let me know who is the transcendental Lord of the universe." He was certain, of course, that not all of them could he the Lord, but the Lord of the universe was one of the three.
(Srimad Bhagavatam------4:1:26-27------translation and purport).

The three deities told Atri Muni: Dear brāhmaṇa, you are perfect in your determination, and therefore as you have decided, so it will happen; it will not happen otherwise. We are all the same person upon whom you were meditating, and therefore we have all come to you.
Atri Muni unspecifically thought of the Personality of Godhead, the Lord of the universe, although he had no clear idea of the Lord of the universe nor of His specific form. Mahā-Viṣṇu, from whose breathing millions of universes emanate and into whom they are again withdrawn, may be accepted as the Lord of the universe. Garbhodakaśāyī Viṣṇu, from whose abdomen sprouted the lotus flower which is the birthplace of Brahmā, may also be considered the Lord of the universe. Similarly, Kṣīrodakaśāyī Viṣṇu, who is the Supersoul of all living entities, may also be considered the Lord of the universe. Then, under the order of Kṣīrodakaśāyī Viṣṇu, the Viṣṇu form within this universe, Lord Brahmā and Lord Śiva may also be accepted as the Lords of the universe.

Viṣṇu is the Lord of the universe because He is its maintainer. Similarly, Brahmā creates the different planetary systems and the population, so he also may be considered the Lord of the universe. Or Lord Śiva, who is ultimately the destroyer of the universe, also may be considered its Lord. Therefore, since Atri Muni did not specifically mention whom he wanted, all three—Brahmā, Viṣṇu and Lord Śiva—came before him. They said, "Since you were thinking of having a son exactly like the Supreme Personality of Godhead, the Lord of the universe, your determination will be fulfilled." In other words, one's determination is fulfilled according to the strength of one's devotion. As stated in Bhagavad-gītā (9.25): yānti deva-vratā devān pitṟn yānti pitṛ-vratāḥ. If one is attached to a particular demigod, one is promoted to the abode of that demigod; if one is attached to the Pitās, or forefathers, one is promoted to their planet; and similarly if one is attached to the Supreme Personality of Godhead, Kṛṣṇa, one is promoted to the abode of Lord Kṛṣṇa. Atri Muni had no clear conception of the Lord of the universe; therefore the three presiding deities who are actually the lords of the universe in the three departments of the modes of nature all came before him. Now, according to the strength of his determination for a son, his desire would be fulfilled by the grace of the Lord.
(Srimad Bhagavatam------4:1:30------translation and purport).

From Atri's tears of jubilation was born a son named Soma, the moon, who was full of soothing rays. Lord Brahmā appointed him the director of the brāhmaṇas, drugs and luminaries.
According to the Vedic description, Soma, the moon-god, was born from the mind of the Supreme Personality of Godhead (candramā manaso jātaḥ). But here we find that Soma was born from the tears in the eyes of Atri. This appears contradictory to the Vedic information, but actually it is not, for this birth of the moon is understood to have taken place in another millennium. When tears appear in the eyes because of jubilation, the tears are soothing. Śrīla Viśvanātha Cakravartī Ṭhākura says, dṛgbhya ānandāśrubhya ata evāmṛtamayaḥ: "Here the word dṛgbhyaḥ means 'from tears of jubilation.' Therefore the moon-god is called amṛtamayaḥ, 'full of soothing rays.' " In the Fourth Canto of Śrīmad-Bhāgavatam (4.1.15) we find this verse:

atreḥ patny anasūyā trīñ
jajñe suyaśasaḥ sutān
dattaṁ durvāsasaṁ somam
ātmeśa-brahma-sambhavān
This verse describes that Anasūyā, the wife of Atri Ṛṣi, bore three sons—Soma, Durvāsā and Dattātreya. It is said that at the time of conception Anasūyā was impregnated by the tears of Atri.

(Srimad Bhagavatam------9:14:3------translation and purport).