This talk presents a preliminary overview of the Islamic tradition of mathematical astronomy based on Ptolemy's Handy Tables and Almagest and sketches some of the developments that took place within this tradition. The overview includes, in particular, the Arabic and Persian commentaries on the Almagest and the category called zij, astronomical handbooks with tables that were influenced by the Almagest in some respects and by the Handy Tables in others. The attempts by Islamic astronomers to modify Ptolemaic astronomy, especially the hay'a tradition and Ibn al-Shatir's New Zij, will be briefly indicated.