An investigation is made into the dynamics involved in the movement of the contact line when one liquid displaces an immiscible second liquid where both are in contact with a smooth solid surface. In order to remove the stress singularity at the contact line, it has been postulated that slip between the liquid and the solid or some other mechanism must occur very close to the contact line. The general procedure for solution is described for a general model for such slip and also for a general geometry of the system. Using matched asymptotic expansions, it is shown that for small capillary number and for small values of the length over which slip occurs, there are either 2 or 3 regions of expansion necessary depending on the limiting process being considered. For the very important situation where 3 regions occur, solutions are obtained from which it is observed that in general there is a maximum value of the capillary number for which the solutions exist. The results obtained are compared with existing theories and experiments.