How did our view of the Universe develop? By the mid-eighteenth century, a world view had developed of a system constrained by physical laws. These laws, if not entirely understood, showed regularity and could be handled mathematically to provide both explanation and prediction of celestial phenomena. Most of us have at least some hazy idea of the fundamental shift that came through the work of Copernicus, Kepler, Galileo and Newton. The idea of a 'Mechanical Universe' running rather like a clock tends to be associated with these sixteenth- and seventeenth-century pioneers. It remains a useful - and perhaps comforting - analogy. Yet, recent investigations based around the Antikythera Mechanism, an artefact from ancient Greece, reinforce a view that the 'Mechanical' conception has been around for a much longer time - indeed certainly as far back as the third century BC. The extent of mechanical design expertise existing around 100 BC as witnessed by the Antikythera Mechanism comes as a great surprise to most people. It is certainly a very ingenious device, often referred to as 'The World's First Computer' although it is really a sophisticated mechanical astronomical calculator with its functions pre-determined rather than programmable. In this review, the structure and functions of the Antikythera Mechanism are described. The astronomy, cosmology and technology inherent in the machine fit surprisingly well into the context of its contemporary Classical world. A strong claim will be made for the influence of such mechanisms on the development of astronomical and philosophical views, based on literary reference. There is evidence that the technology persisted until its spectacular and rather sudden re-appearance in Western Europe around 1300 AD. From then on it is not hard to chart a path through the astronomical clocks of the sixteenth century to Kepler's aim (expressed in a 1605 letter) to 'show that the heavenly machine is not a kind of divine, live being, but a kind of clockwork …', and on to the widespread development of popular visualisation of the heliocentric Solar System in the orreries of the eighteenth century.