  In this paper we give a survey of elliptic theory for operators associated
with diffeomorphisms of smooth manifolds. Such operators appear naturally in
analysis, geometry and mathematical physics. We survey classical results as
well as results obtained recently. The paper consists of an introduction and
three sections. In the introduction we give a general overview of the area of
research. For the reader's convenience here we tried to keep special
terminology to a minimum. In the remaining sections we give detailed
formulations of the most important results mentioned in the introduction.
