The theory of eigenvalues and eigenvectors of rectangular matrix pencils is complicated by the fact that arbitrarily small perturbations of the pencil can cause them disappear. However, there are applications in which the properties of the pencil ensure the existence of eigenvalues and eigenvectors. In this paper it is shown how to develop a perturbation theory for such pencils. (Also cross-referenced as UMIACS-TR-91-105)