Zel'dovich mapping can be characterized in terms of the caustics that form in the density field. Different types of caustics can be classified using catastrophe theory, at least for Zel'dovich mapping in two dimensions. We have implemented this scheme numerically so that the caustic structure in a given case can be determined using the initial gravitational potential. We find that the higher order caustics cluster strongly, and are generally found to coincide with large clusters. Gravitational lensing by a thin lens has the same kind of mapping as the Zel'dovich mapping in two dimensions. We exploit this fact to describe gravitational lenses in the same language and suggest a new representation for lens models.