Anātman (Sanskrit: अनात्मन्) in Sanskrit means not spiritual, corporeal, unreal, something different from spirit or soul, not self, another. In Advaita Vedanta this word is used to indicate Samsara i.e. the world - existence or the world phenomena, which is the unreal projected by ignorance on the real, that is, on Brahman who is the Absolute. In Buddhism as Anatta it indicates Not-self and the illusion of self. Nāgārjuna expounded the Buddhist theory of non-substantiality (anatman) or emptiness (sunyata), and Adi Shankara explained the concept of Anatman in the light of Upanishadic Thought; the exact time when these two philosophers lived is still debated.
Upanishadic concept of the Atman
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A proper understanding of the word, Anātman, cannot be had without the proper understanding of the Upanishadic concept of the Atman. The Vedic seers had long ago realised that the Atman is Brahman. They had realised the universal truth – ayam atman Brahman - that the Jiva, the individual soul, is not merely a part of Brahman, the Universal or Cosmic Soul, the Jiva is Brahman itself. Therefore, Adi Shankara in his Vivekachudamani explains - “I am the body”, thus thinks a foolish man. A man of mere book-knowledge considers himself to be a combination of the body and the Jiva. But the realised sage, because of his discrimination, knows that “I am Brahman”, and looks upon the eternal Atman as his Self, And, explaining thus, instructs - “Just as you would not identify yourself with your shadow, your reflection, your dream-body or the body in your heart’s imagination, so too, you should not identify yourself with your living body.”  Also meaning thereby that all human beings can aspire to be liberated, and Self-realisation is possible during one’s own life-span; one need not wait for death to achieve this objective.
According to the Upanishads the world phenomena is a creation of Brahman, a manifestation of Brahman who after creating the world created first the intermediary Virat subsisting between Itself and the Universe, created the couple Rayim and Prana i.e. Food and Prana, to produce in multifarious ways creatures that are impelled by Consciousness.
Sandilya, the metaphysical philosopher of the Chandogya Upanishad, provides to us the cosmological proof of the Absolute which he calls Tajjalan – Sarvam khaluidam Brahman tajjalaaniti shaant upaasite – “All this (collectively) is Brahman, indeed: what evolves from That, what dissolves in That, what breathes or functions in That, should be closely and calmly studied…….”  Thus, to the Vedic seers, Brahman, having all consciousness factors and all non-consciousness factors in abundance, appears as the mind, has prana as the body, and has the form of consciousness; constitutionally Brahman is – Mind, Life, Light (that which illuminates or reveals), Prana-sarira, Satya-sankalpah, etc. From Brahman emanate all organs, all worlds, all gods and all beings. Therefore, cosmologically, Brahman is the subtle essence that underlies phenomena, biologically, it is the supreme life-principle that gives life to the universe and psychologically, it annihilates all individualities. As Ishwara Brahman controls the sentient and the insentient, both. The world exists in time and space governed by causality.
Because volition exercising Consciousness and its creation are both identical and there is one non-dual Pure Consciousness only, Adi Shankara concludes that this world of manifoldness is unreal i.e. Mithya, which is the wrong attribution of the qualities of the non-consciousness factor to the consciousness factor. This false attributing of the true concept of one to that of another i.e. false super-imposition, which is an eternal process, is called Adhyasa by Adi Shankara. This super-imposition is called Ignorance. But, Ignorance is not want of knowledge. Super-imposition is the apparent presentation to consciousness, by way of remembrance, of something previously observed in some other thing; it is illusory knowledge. Texts are based on this false super-imposition. Direct perception as well as the Shastras are part of Avidya, the world of duality fashioned by Maya. The object of Vedanta is to clarify the misconception that direct perception and Maya are real. The knower who has attained self-identification with the body, mind and senses etc., for him all means of knowledge become irrelevant. From the standpoint of the Absolute Adi Shankara regards the world as a false appearance while recognizing its empirical reality and denying its ontological reality.
Buddhist concept of Anatman or Anatta
In Buddhism, Anatman i.e. Anatta is Not-self or illusion of self. The Buddhists believe that there is no permanent underlying substance in human beings called the soul. They believe that non-existence of soul (anatta/anatman), the impermanence of beings (anicca/anitya) and suffering (dukkha) are the three characteristics (tri-lakkhana) of all existence, and understanding of these three constitutes right understanding. But the Buddhists did not coin this word, this word existed and was in use before Buddha’s time, even the Sanskrit lexicons and the Pali lexicons while providing the meaning of Anatman or Anatta, both, do not suggest that Anatman is a doctrine for Buddha never taught either “atman” or “anatman”, and never did Buddhism teach that “anatman” exists. According to Wayman Nāgārjuna must have heard both being referred to as doctrines and wrote verse XVII.6 – …(to some) the existence of the Self is taught..(to others) the non-existence of the Self is taught.. To those endowed with receptivity (ksanti) for the vast and profound doctrine (dharma), it is taught that neither the Self nor the Non-self exists. in his Madhyamakakarika that gave the general impression that this was what Buddha had actually taught. Buddha sought to eliminate the concept of the atman and not of the Atman. Upanishads sought to free individuals from ego-attachment by pointing out that the real self is the Universal Self rather than the individual self, the Buddha sought to free individuals from ego-attachment by pointing out that there is no individual self to which to become attached. The theory of non-substantiality (anatman) or emptiness (sunyata) that Nāgārjuna had attempted to explain was not palatable to the substantialist philosophers who raised the objection discussed in the first six verses of Chapter 24 of his Mulamadhyamakakarika on Truth.
Advaita concept of Anatman
The Atman is the substratum of knowing, doing and enjoying owing to the superimposition of the unreal i.e. Anatman, on the real i.e. Atman, there is no opposition between these two because they are non-different.Existence is the common factor in both, conscious existence in the Absolute and non-conscious existence in the Absolute, and both are eternal. Thus, Man resembles both Brahman and the universe. Adi Shankara calls the conscious factor Atman and the non-conscious factor as the Anatman. He does not use the term Anatman to mean non-soul or anti-soul. The Atman is formless and partless whose true nature cannot be perceived but whose true nature alone can be experienced but the Anatman causes the living being to experience the pangs of (repeated) birth, sufferings, old age and death. Anatman, a creation of Brahman which is non-different from Brahman and has no existence apart from Brahman, illumined by the light of consciousness is Mithya, deemed eternal because it subsists projected (by the mind) upon Brahman acting as the substratum. The instruction is to discriminate Atman and Anatman to become liberated. Suresvara in terms of causality admits ignorance to be the source for the relation which the atman has with what is ultimately unreal and superimposed upon it…..The very nature of anatman is ignorance, it is completely dependent on ignorance and has no nature of its own apart from it. Therefore, logically beside the sole non-dual real entity that the Atman is no unreal thing such as the Anatman can ever exist.
In order to realise the self-existent eternal Atman, the seeker after Truth is required to separate it from the Anatman, he is required to identify the Atman by negation of the Anatman. Therefore, such a seeker is entitled to know about the Anatman too. ko’sau anatman? The Anatman comprises the sthulasarira, the physical body, the suksmasarira, the subtle body and the karanasarira, the causal body, which are the cause of bondage owing to a) the avaranasakti, a tamasic component, which conceals the real nature of the Atman, and b) viksepasakti resulting from its rajasic component which projects the non-atman as the Atman. After knowing this one proceeds to determine the nature of Brahman. When nirguna Brahman is conceived as saguna, with the power to start the cosmic process with the aid of maya, it is known as Ishvara, who in essence is not different from the nirguna Brahman, the difference between the two Brahmans is only in the manner of conception and approach, whether by way of jnana to understand and realise the Supreme Principle of the Universe or by way of bhakti to realise the same Principle as one responsible for the creation of the world. 
- Swami Madhavananda. Vivekachudamani of Adi Shankara. Kolkata: Advaita Ashrama. p. 12. "Verse 20"
- Swami Madhavananda. Brihadaranyaka Upanishad. Kolkata: Advaita Ashrama. p. 202. "Verse 4.4.5"
- Swami Chinmayananda. Vivekachoodamani of Adi Shankara. Chinmaya Mission. p. 213. "Verse 160"
- Swami Chinmayananda. Vivekachoodamani of Adi Shankara. Chinmaya Mission. p. 216. "Verse 163"
- Swami Gambhirananda. Eight Upanishads Vol I. Kolkata: Advaita Ashrama. p. 20. "Aitareya Upanishad. 1.1.1"
- Swami Gambhirananda. Eight Upanishads Vol II. Kolkata: Advaita Ashrama. p. 81. "Mundaka Upanishad. 1.1.1"
- Swami Gambhirananda. Eight Upanishads Vol I. Kolkata: Advaita Ashrama. p. 25. "Aitareya Upanishad. 1.1.4"
- Swami Gambhirananda. Eight Upanishads Vol I. Kolkata: Advaita Ashrama. pp. 411–2,423. "Prashna Upanishad. 1.4-5, 15"
- Swami Gambhirananda. Eight Upanishads Vol I. Kolkata: Advaita Ashrama. p. 71. "Aitareya Upanishad. 3.1.3"
- Swami Gambhirananda. Chandogya Upanishad. Kolkata: Advaita Ashrama. p. 208. "Verse 14.1"
- Swami Gambhirananda. Chandogya Upanishad. Kolkata: Advaita Ashrama. p. 210. "Verse 14.2"
- Swami Madhavananda. Brihadaranyaka Upanishad. Kolkata: Advaita Ashrama. p. 202. "Verse 2.2.20"
- Swami Gambhirananda. Chandogya Upanishad. Kolkata: Advaita Ashrama. pp. 167–190. "Verse 3.1-11"
- Swami Madhavananda. Brihadaranyaka Upanishad. Kolkata: Advaita Ashrama. p. 343. "Verse 3.6.1"
- Swami Gambhirananda. Shvetashvatara Upanishad. Kolkata: Advaita Ashrama. pp. 59, 65. "Verse 1.4-5"
- Swami Gambhirananda. Eight Upanishads Vol I. Kolkata: Advaita Ashrama. p. 189. "Katha Upanishad. 2.2.5"
- Swami Jagadananda. Upadesasahasri of Adi Shankara. Chennai: Sri Ramakrishna Math. p. 140. "Verse 14.44"
- Swami Vireshwarananda. Brahma Sutras of Badarayana. Kolkata: Advaita Ashrama. p. 7. "Adhyasa or super-imposition"
- D.Cox & others. Theology of Religions:Christianity and Other religions. p. 342.
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- David J. Kalupahana. Mulamadhyamakakarika of Nagarjuna: the philosophy of the middle way. New Delhi: Motilal Banarsidass Publishers. p. 67.
- Swami Madhavananda. Vivekachudamani of Adi Shankara. Kolkata: Advaita Ashrama. pp. 46–51. "Verse 125-138"
- Swami Madhavananda. Vivekachudamani of Adi Shankara. Kolkata: Advaita Ashrama. pp. 45–46. "Verse 124-125"
- Swami Madhavananda. Vivekachudamani of Adi Shankara. Kolkata: Advaita Ashrama. p. 56. "Verse 148"
- Swami Madhavananda. Vivekachudamani of Adi Shankara. Kolkata: Advaita Ashrama. p. 12. "Verse 20"
- Swami Madhavananda. Vivekachudamani of Adi Shankara. Kolkata: Advaita Ashrama. p. 58. "Verse 152"
- Ivan Kocmarek. Language and Release: Sarvajnatman’s Pancaprakriya. New Delhi: Motilal Banarsidass Publishers. p. 39.
- Sri Chandrashekhara Bharati III of Sringeri. Sri Samkara’s Vivekacudamani. Mumbai: Bharatiya Vidya Bhavan. p. xxi,xxix.