We will present a number of mathematical models for quasi-static or dynamic contact of rods and beams. In these models the contact is either with friction (modeled by versions of Coulomb's law) or frictionless, and the setting is either isothermal or includes thermal effects. Recently we constructed and analyzed a number of such models. The aim is to gain insight and understanding into the evolution of the process of contact in settings that avoid some of the mathematical difficulties that are related to two or three dimensions. Then we discuss and present some numerics. The first part of the talk will be devoted to the models and will deal with some of the mathematical difficulties. In the second part of the talk we will present some of our numerical algorithms for these problems, explain some of our numerical difficulties, and present a number of results of numerical simulations. We will concentrate more on what we just started computing numerically and suggest a number of interesting problems that need numerical analysis but, even more importantly, reliable numerical algorithms and simulations.