Real Abbasid power lasted about a century. From the ninth century provinces broke away under their own dynasties, rival caliphates were proclaimed by the Fatimids in Cairo and the Umayyads in Cordoba, and from 945 the caliph in Baghdad was under the control of the Buyid amirs and from 1055 of the Seljuk sultans. What he retained was legitimacy: a sultan who could not claim the caliph's investiture ruled without a licence, and the office survived on that function alone for three hundred years.
It ended in 1258, when Hulagu's Mongol army took Baghdad, destroyed the libraries and killed the caliph al-Mustasim, by tradition rolled in a carpet and trampled so that no royal blood should touch the ground. The Mamluk sultans of Egypt installed a surviving Abbasid in Cairo in 1261, and that shadow line performed investitures without power until the Ottoman conquest of 1517.
Mona Hassan's history follows what the loss meant to Muslims who lived through it and to those who invoked it afterwards. The caliphate was mourned in poetry and legal argument within a generation of 1258, reconstructed as a claim by later empires, and mourned again after 1924, and the vacancy it left continues to organise a great deal of modern Islamic political thought.