When, exactly, was it written?
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http://www.veda.harekrsna.cz/encyclopedia/sb.htm
13th century CE as a date for Srimad Bhagavatam was accepted by scholars until the half of 20th century. Nevertheless, the work Tahqiq-i-Hind by Alberuni, a Muslim who made a study of the India in the 10th century CE, mentions Bhagavata Purana.
Nowadays the date 9th century CE predominates. The critics claim that the Bhagavatam mentions Hunas and thus it had to be written after their invasion to India in 5th century CE. Nevertheless, the Hunas are mentioned also in older texts such as Lalitavistara, a Buddhist work dated to 3rd century CE, and Mahabharata, a recognized archaic work. These evidences indicate that this race was known before the invasion and its mention in the Bhagavatam doesn't suggest any occupation.
Others argue that Bhagavatam was influenced by the philosopher Sankara (commonly accepted: 8th century CE) because of similarity of ideas and of the language. But Gaudapada, teacher of Govinda who was teacher of Sankara, mentions the Bhagavatam in his Uttaragita-bhasya and in his commentary on Sankhya-karika makes reference to other two verses.