The character and content of Babylonian scientific or mathematical astronomy, as we know it from texts of the last half millennium B.C., are sketched. This late-Babylonian astronomy is set in contrast to earlier Babylonian astronomy as well as to the kinds of astronomy found in other ancient cultures, and an attempt is made at a very broad classification of such pre-scientific astronomies. The lateness and uniqueness of Babylonian mathematical astronomy is emphasized, and it is shown that its creation depended upon the availability of a peculiar set of ingredients, e.g., a particular type of mathematics, and a tradition of making and recording observations of certain astronomical phenomena. It is finally argued that all subsequent varieties of scientific astronomy, in the Hellenistic world, in India, in Islam, and in the West - if not indeed all subsequent endeavour in the exact sciences - depend upon Babylonian astronomy in decisive and fundamental ways.