 Our survey begins in section   1, which unpacks some meanings of ‘infinite’, and traces various philosophical conceptions of infinity from ancient times to the 19th Century. Section   2 turns to the historical development of the mathematics of infinity over a comparable period. This provides background to a presentation in section   3 of modern mathematics’ treatment of infinity—some infinite number systems, infinities of measure, of counting, of calculus, and infinitary operations on numbers. This in turn sets the stage for our discussion of mathematical ontology in section   4.