Here's how it appears frequently in most Euro-centric historical accounts: the library was first gutted in 47AD during Julius Caesar's war with the contested Rulers of Egypt, the library managed a comeback; that is until the Caliph Omar (re)captured Alexandria in 640AD. Upon taking the city after a prolonged and frustrating siege, he is said to have had the entire 200,000 volumes removed, with the justification "If these writings of the Greeks agree with the Book of God, they are redundant and need not be preserved; if they disagree, they are pernicious things and ought to be destroyed." The books were at least 'recycled'; they were used to heat the bathhouses of the city for six months. The actual fate of the library and its texts is fairly unclear. Most serious scholars seem to think the scrolls were slowly superseded by the more convenient and portable codex book, and that the older materials were slowly distributed and dispersed to other centers of learning around the Mediterranean and Middle East.
    Sooooo... this was pretty much the line taken throughout the Renaissance and much of the Enlightenment, the reason being that historians, scholars and poets generally drooled over the purloined, much-idealized contents of the Museion for centuries afterwards. The Europeans, by the 12-13th centuries, were acutely aware of the fact the written tradition of Antiquity had slipped from their grasp, and they needed someone to blame. The Crusades were well under way, so who better to point the finger at than the nasty infidel Arabs.
    However....modern scholarship in comparative historiography, textual analysis, archeology and codiocology indicate the library was likely looted or damaged on numerous occasions following the height of its collection and standards in the 2-3rd c. BC, with the worst damage likely inflicted by rioting Christians as they protested Roman suppression just prior to emperor Constantine's conversion. The actual story of its destruction at the hands of Omar, the head of the Arab army in the 8th c., may have been cultivated by sectarian historians in Baghdad much later, wishing to paint the Ummyad Caliph of the time as a fanatic. This was later picked up by European sources, who found it colorful and politically expedient.
On the Library of Alexandria and its fate under Moslem rule

Canfora, Luciano. The vanished library translated by Martin Ryle. Berkeley : University of California Press, 1989.

Blum, Rudolf, 1909- Kallimachos : the Alexandrian Library and the origins of bibliography translated from the German by Hans H. Wellisch. Madison, Wis. : University of Wisconsin Press, c1991. 282p.

El-Abbadi, Mostafa. The life and fate of the ancient Library of Alexandria Paris : UNESCO,1990. 250p. See in particular, p.88- 102, on the mingling of Greek, Roman, Perisan and Arabic materials, as well as a close examination of the historical sources discussing the collection's eventual dispersal and destruction.

Epistola ad Philocratem. The letter of Aristeas, London : Society for promoting Christian knowledge. New York : The Macmillan co., 1918.

Fraser, P. M. Ptolemaic Alexandria. Oxford, Clarendon Press, 1972. 3 v.

Parsons, E. A. The Alexandrian library; glory of the Hellenic world, its rise, antiquities, and destruction. -- Amsterdam : American Elsevier Press, 1952.

Theim, Jon. "The Great Library of Alexandria Burnt : Towards the History of a Symbol," Journal of the History of Ideas, 40, 4 (1979), 507 -26.

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