Many MEMS devices employ parallel plates for capacitive sensing and actuation. The desire to get a significant change in capacitance has been pushing the need to reduce the gap between the moving plate and the fixed plate. With fabrication processes making rapid strides, it is now possible to push the gap to be so small that it becomes comparable to the mean free path of gas or air molecules present in the gap. In all MEMS devices, where the essential function of the device depends on the dynamics of the mechanical components, the presence of air or a gas in such narrow gaps leads to energy dissipation if the gas is squeezed between the two plates due to transverse motion of the movable plate. This energy dissipation, known as squeeze film damping, plays a critical role in determining the quality factor of such devices. For many devices, simple approximation of squeeze film damping based on the linearized Reynolds equation is sufficient. However, under moderate vacuum and very narrow gaps, the linearized Reynolds equation does not give satisfactory results, especially if large amplitude motions of the movable plate are desired. In this paper, we carry out an analysis of the fluid flow in the narrow gap taking rarefaction and surface roughness into account and show that both these factors have a significant effect on the squeeze film damping of the devices.