According to Wikipedia:-
Whilst the Bodleian Library, in its current incarnation, has a continuous history dating back to 1602, its roots date back even further. The first purpose-built library known to have existed in Oxford was founded in the fourteenth century by Thomas Cobham, Bishop of Worcester. This small collection of chained books was situated above the north side of the University Church of St Mary the Virgin on the High Street. This collection continued to grow steadily, but when, between 1435 and 1437 Humphrey, Duke of Gloucester (brother of Henry V of England), donated a great collection of manuscripts, the space was deemed insufficient and a larger building was required. A suitable room was finally built above the Divinity School, and completed in 1488. This room continues to be known as Duke Humfrey’s Library.