The Library of Alexandria caught fire several times.
The first may have been when the Romans conquered Egypt. The Romans burned their own ships and much of the city caught fire, and the library may have been partially destroyed at this time.
A branch of the library may have been burned with the destruction of pagan temples when the Roman Empire outlawed paganism, but nobody knows how many (if any) books were lost. The main building was apparently not affected. And by the time paganism was made illegal in the Roman Empire, a concerted effort had been made to have copies of important documents in other libraries, including the worlds largest library at Constantinople. These other libraries were not burned (though it's entirely possible that some books in them were destroyed).
And it was finally destroyed by the Muslim army. There is a story that the Caliph ordered the burning of books stating that if they contradicted the Quran they are heretical, and if they did not then they are redundant. There are no contemporary sources for this story, so most historians doubt it. Whether or not this burning was deliberate, the destruction was complete and library was lost to history.