Repetitive processes are a distinct class of 2D systems of both practical and theoretical interest. Their essential characteristic is repeated sweeps, termed passes, through a set of dynamics defined over a finite duration with explicit interaction between the outputs, or pass profiles, produced as the process evolves. Experience has shown that these processes cannot be studied/controlled by direct application of existing theory (in all but a few very restrictive special cases). This fact, and the growing list of applications areas, has prompted an on-going research programme into the development of a ‘mature’ systems theory for these processes for onward translation into reliable generally applicable controller design algorithms. It has long been considered that Volterra operator techniques should have a key role to play in this general area. In this paper, we first present the necessary properties of a Volterra operator representation for the very important sub-class of so-called discrete linear repetitive processes and then use them to develop a characterization of stability in this setting.