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Squeeze film effects naturally occur in dynamic MEMS structures because most of these structures employ parallel plates or beams that trap a very thin film of air or some other gas between the structure and the fixed substrate. An accurate estimate of the effect of squeeze film is important for predicting the dynamic performance of such devices. In design, availability of very good models for squeeze film effects is indispensable. In this paper, we discuss the development of squeeze film flow modelling, tracking its routes to the air damped vibrating system studies in the early twentieth century. We try to capture the early developments in gas lubrication theory and then discuss the current developments motivated by the complexities in squeeze film flow analysis brought out by the geometries and flow conditions prevalent in dynamic MEMS devices.