The Two-Sided Perfect Matching model is generalised to an Imperfect Matching model with search frictions. A search model is proposed which is characterised by bilateral search and vertical heterogeneity and allows for a generally specified utility function. The fundamental result is that with common beliefs about the evolving population, a unique Imperfect Matching equilibrium exists in iterated strict dominance. This holds independent of the characteristics of the utility function. The properties of equilibrium are analysed for different pay-off specifications. It is shown that for multiplicatively separable pay-offs both Steady State distributions are endogenously partitioned into classes. This fails to hold however out of Steady State. With search frictions disappearing, the limit case of the search model is the Gale-Shapley-Becker Perfect Matching model.