As the supreme controller of both the material and spiritual worlds, the Lord has different incarnations of unlimited categories. Incarnations like Brahmā, Rudra,Manu, Pṛthu and Vyāsa are His material qualitative incarnations, but His incarnations like Rāma, Narasiṁha, Varāha and Vāmana are His transcendental incarnations. Lord Śrī Kṛṣṇa is the fountainhead of all incarnations, and He is therefore the cause of all causes.
(Srimad Bhagavatam------1:1:17------purport).

The list of incarnations of the Personality of Godhead given herein is not complete. It is only a partial view of all the incarnations. There are many others, such as Śrī Hayagrīva, Hari,Haṁsa, Pṛśnigarbha, Vibhu, Satyasena, Vaikuṇṭha, Sārvabhauma, Viṣvaksena, Dharmasetu, Sudhāmā, Yogeśvara, Bṛhadbhānu and others of the bygone ages. Śrī Prahlāda Mahārāja said in his prayer, "My Lord, You manifest as many incarnations as there are species of life, namely the aquatics, the vegetables, the reptiles, the birds, the beasts, the men, the demigods, etc., just for the maintenance of the faithful and the annihilation of the unfaithful. You advent Yourself in this way in accordance with the necessity of the different yugas. In the Kali-yuga You have incarnated garbed as a devotee." This incarnation of the Lord in the Kali-yuga is Lord Caitanya Mahāprabhu. There are many other places, both in the Bhāgavatam and in other scriptures, in which the incarnation of the Lord as Śrī Caitanya Mahāprabhu is explicitly mentioned. In the Brahma-saṁhitā also it is said indirectly that although there are many incarnations of the Lord, such as Rāma, Nṛsiṁha, Varāha, Matsya, Kūrma and many others, the Lord Himself sometimes incarnates in person. Lord Kṛṣṇa and Lord Śrī Caitanya Mahāprabhu are not, therefore, incarnations, but the original source of all other incarnations. This will be clearly explained in the next śloka. So the Lord is the inexhaustible source for innumerable incarnations which are not always mentioned. But such incarnations are distinguished by specific extraordinary feats which are impossible to be performed by any living being. That is the general test to identify an incarnation of the Lord, directly and indirectly empowered. Some incarnations mentioned above are almost plenary portions. For instance, the Kumāras are empowered with transcendental knowledge. Śrī Nārada is empowered with devotional service. Mahārāja Pṛthu is an empowered incarnation with executive function. The Matsya incarnation is directly a plenary portion. So the innumerable incarnations of the Lord are manifested all over the universes constantly, without cessation, as water flows constantly from waterfalls.
(Srimad Bhagavatam------1:3:26------purport).

Lord Śrī Caitanya Mahāprabhu declared that Śrīmad-Bhāgavatam is the spotless sound representation of all Vedic knowledge and history. There are selected histories of great devotees who are in direct contact with the Personality of Godhead. Śrīmad-Bhāgavatam is the literary incarnation of Lord Śrī Kṛṣṇa and is therefore nondifferent from Him. Śrīmad-Bhāgavatam should be worshiped as respectfully as we worship the Lord. Thereby we can derive the ultimate blessings of the Lord through its careful and patient study. As God is all light, all bliss and all perfection, so also is Śrīmad-Bhāgavatam.
(Srimad Bhagavatam------1:3:40------purport).

The sound incarnation of Lord Kṛṣṇa, the Supreme Soul (i.e. Śrīmad-Bhāgavatam), enters into the heart of a self-realized devotee, sits on the lotus flower of his loving relationship, and thus cleanses the dust of material association, such as lust, anger and hankering. Thus it acts like autumnal rains upon pools of muddy water.
(Srimad Bhagavatam------2:8:5------translation).

The Lord's incarnations in different species of life are all transcendental. He appears as a human being in His incarnations of Kṛṣṇa, Rāma, etc., but He is not a human being. Anyone who mistakes Him for an ordinary human being is certainly not very intelligent, as confirmed in Bhagavad-gītā (B.G.9:11): avajānanti māṁ mūḍhā mānuṣīṁ tanum āśritam. The same principle is applicable when He appears as the hog or fish incarnations. They are transcendental forms of the Lord and are manifested under certain necessities of His own pleasure and pastimes. Such manifestations of the transcendental forms of the Lord are accepted by Him mostly to enliven His devotees. All His incarnations are manifested whenever there is a need to deliver His devotees and maintain His own principles.
(Srimad Bhagavatam------3:9:19------purport).

Pṛthu Mahārāja was a powerful incarnation of Lord Kṛṣṇa's potencies; consequently any narration concerning his activities is surely very pleasing to hear, and it produces all good fortune. As far as I am concerned, I am always your devotee as well as a devotee of the Lord, who is known as Adhokṣaja. Please therefore narrate all the stories of King Pṛthu, who, in the form of the son of King Vena, milked the cow-shaped earth.
Lord Kṛṣṇa is also known as avatārī, which means, "one from whom all the incarnations emanate." In Bhagavad-gītā (B.G.10:8) Lord Kṛṣṇa says, ahaṁ sarvasya prabhavo mattaḥ sarvaṁ pravartate: "I am the source of all spiritual and material worlds. Everything emanates from Me." Thus Lord Kṛṣṇa is the origin of everyone's appearance. As far as this material world is concerned, Lord Brahmā, Lord Viṣṇu and Lord Śiva are all emanations from Kṛṣṇa. These three incarnations of Kṛṣṇa are called guṇa-avatāras. The material world is governed by three material modes of nature, and Lord Viṣṇu, Lord Brahmā and Lord Śiva respectively take charge of the modes of goodness, passion and ignorance. Mahārāja Pṛthu is also an incarnation of those qualities of Lord Kṛṣṇa by which one rules over conditioned souls.
(Srimad Bhagavatam------4:17:6-7------translation and purport).

The activities of the Supreme Personality of Godhead in His various forms and incarnations are always uncommon and wonderful. It is not possible for a tiny human being to estimate the purpose and plans of such activities; therefore Śrīla Jīva Gosvāmī has said that unless the Lord's activities are accepted as inconceivable, they cannot be explained. The Lord is eternally existing as Kṛṣṇa, the Supreme Personality of Godhead, in Goloka Vṛndāvana. He has also simultaneously expanded Himself in innumerable forms, beginning with Lord Rāma, Lord Nṛsiṁha, Lord Varāha and all the incarnations coming directly from Saṅkarṣaṇa. Saṅkarṣaṇa is the expansion of Baladeva, and Baladeva is the first manifestation of Kṛṣṇa. Therefore all these incarnations are known as kalā.
The word īśvarāṇām refers to all the Personalities of Godhead. As stated in Brahma-saṁhitā (5.39): rāmādi-mūrtiṣu kalā-niyamena tiṣṭhan. In the Śrīmad-Bhāgavatam it is confirmed that all the incarnations are partial expansions, or kalā, of the Supreme Personality of Godhead. However, Kṛṣṇa is the original Supreme Personality of Godhead. One should not take the plural number of the word īśvarāṇām to mean that there are many Godheads. The fact is that God is one, but He exists eternally and expands Himself in innumerable forms and acts in various ways. Sometimes the common man is bewildered by all this and considers such activities contradictory, but they are not contradictory. There is a great plan behind all the Lord's activities.
(Srimad Bhagavatam------4:17:36------purport).

By the evidence of śāstra, Sanātana Gosvāmī detected Lord Caitanya Mahāprabhu to be a direct incarnation of Kṛṣṇa, the Supreme Personality of Godhead, although Lord Caitanya never disclosed the fact. It is therefore generally recommended that the ācārya, or guru, should not be accepted as an ordinary man.
(Srimad Bhagavatam------4:21:11------purport).

Śrī Caitanya Mahāprabhu has said, āpani ācari prabhu jīvere śikhāya. It is very well known that whatever Śrī Caitanya Mahāprabhu taught in His life as ācārya, He Himself practiced. When He was preaching as a devotee, although He was detected by several great personalities to be the incarnation of Kṛṣṇa, He never agreed to be addressed as an incarnation. Even though one may be an incarnation of Kṛṣṇa, or especially empowered by Him, he should not advertise that he is an incarnation.
(Srimad Bhagavatam------4:22:5------purport).

The word manave, meaning "unto the supreme Manu," is also significant. The supreme Manu in Vedic literature is Svāyambhuva Manu, who is an incarnation of Kṛṣṇa. All the Manus are empowered incarnations of Kṛṣṇa (manvantara-avatāra).
(Srimad Bhagavatam------4:24:42------purport).

The sāṅkhya-yoga system popular at the present moment was propounded by the atheist Kapila, but the original sāṅkhya-yoga system was propounded by anincarnation of Kṛṣṇa also named Kapila, the son of Devahūti. Similarly, Dattātreya, another incarnation of Kṛṣṇa, also explained the sāṅkhya-yoga system. Thus Kṛṣṇa is the origin of all sāṅkhya-yoga systems and mystic yoga powers.
(Srimad Bhagavatam------4:24:42------purport).

Kṛṣṇa incarnated Himself as Lord Buddha to put an end to animal-killing in the name of religion. Animal sacrifice under the name of religion is conducted by the influence of tamo-guṇa (the mode of ignorance).
(Srimad Bhagavatam------4:27:11------purport).

Lord Buddha, an incarnation of Lord Kṛṣṇa, adopted a particular means to propagate the philosophy of bhāgavata-dharma. He preached almost exclusively among atheists. Atheists do not want any God, and Lord Buddha therefore said that there is no God, but he adopted the means to instruct his followers for their benefit. Therefore he preached in a duplicitous way, saying that there is no God. Nonetheless, he himself was an incarnation of God.
(Srimad Bhagavatam------5:15:1------purport).

Soon after the cosmic creation, the entire universe was inundated with water. At that time Lord Kṛṣṇa (Keśava) incarnated as a gigantic fish to protect the Vedas. Therefore Manu addresses Lord Matsya as mukhyatama, the first incarnation to appear. Fish are generally considered a mixture of the modes of ignorance and passion, but we must understand that every incarnation of the Supreme Personality of Godhead is completely transcendental. There is never any deterioration of the Supreme Lord's original transcendental quality. Therefore the word sattvāya is used here, meaning pure goodness on the transcendental platform. There are many incarnations of the Supreme Lord: Varāha mūrti (the boar form), Kūrma mūrti (the tortoise form), Hayagrīva mūrti (the form of a horse) and so on. Yet we should never think any of Them material. They are always situated on the platform of śuddha-sattva, pure transcendence.
(Srimad Bhagavatam------5:18:25------purport).

In this age of Kali, however, there is a great concession given to people in general. Yajñaiḥ saṅkīrtana-prāyair yajanti hi sumedhasaḥ: (S,B.11:5:32) we may worship Lord Caitanya, the hidden incarnation of Kṛṣṇa. Kṛṣṇa-varṇaṁ tviṣākṛṣṇam: although He is Kṛṣṇa Himself, He always chants Hare Kṛṣṇa and preaches Kṛṣṇa consciousness. One is recommended to worship this incarnation by chanting, the saṅkīrtana-yajña. The performance of saṅkīrtana-yajña is a special concession for human society to save people from being affected by known or unknown sinful activities. We are surrounded by unlimited sins, and therefore it is compulsory that one take to Kṛṣṇa consciousness and chant the Hare Kṛṣṇa mahā-mantra.
(Srimad Bhagavatam------9:16:23------purport).

The fifty-fifth verse says, svayam eva hariḥ kila, indicating that Kṛṣṇa, the eighth son of Devakī, is the Supreme Personality of Godhead. Kṛṣṇa is not an incarnation. Although there is no difference between the Supreme Personality of Godhead Hari and His incarnation, Kṛṣṇa is the original Supreme Person, the complete Godhead. Incarnations exhibit only a certain percentage of the potencies of Godhead; the complete Godhead is Kṛṣṇa Himself, who appeared as the eighth son of Devakī.
(Srimad Bhagavatam------9:24:53-55------purport).

The Hare Kṛṣṇa movement is also an incarnation of Kṛṣṇa in the form of the holy name (nāma-rūpe). Every one of us who is actually afraid of the asuric rulers and politicians must welcome this incarnation of Kṛṣṇa: Hare Kṛṣṇa, Hare Kṛṣṇa, Kṛṣṇa Kṛṣṇa, Hare Hare/ Hare Rāma, Hare Rāma, Rāma Rāma, Hare Hare.
(Srimad Bhagavatam------10:3:21------purport).