We begin with an account of the Prime Mover argument. This originated in the "laws" of Plato, where it is argued that a self-moving mover must exist as source of other motions, and that it must be a kind of soul. In Aristotle this Prime Mover is not itself moved, and elaborate proofs of its existence are offered in the "Physics": all motion requires a mover, and the series cannot go on to infinity, but must end in one or more unmoved movers. His proofs, however, were far from watertight, and later Peripatetics like Theophrastus and Strato rejected them. The argument reappeared in Proclus, but only as subsidiary to a First Cause argument; moreover,the Prime Mover is only the second member of the Neoplatonic "trinity". John Philoponus' theory of "impetus" should have undermined the argument, but in fact did not, and it continued to be used by the Arabs (despite criticism by Avicenna) and the Jews (notably Maimonides). It was taken over by the Christian scholastics like Aquinas. But criticism also continued, especially from Algaael in Islam and by Ockham and his followers in Christendom, and a detailed refutation was offered by the Jew Crescas. The arrival of non-Aristotelian physics was fatal to the argument; it is indeed still defended occasionally by neo-Scholastic philosophers, but none of their defences is adequate. Outside Scholasticism it has few supporters, though Samuel Clarke used it for the Platonic purpose of pointing out an analogy between the Prime Mover and mind, Lotze, however, advanced a quite different kind of argument, but based, like the Platonic and Aristotelian ones, on the existence of change; he argued that change ought always to be internal to that which changes, and hence that the universe must be in some sense a unity. The relationship between this unity and individual things would then be analogous to that between a mind and its states. [Continued in text ...]