This chapter considers the fundamentals of axially moving materials. We systematically develop and solve a simplified model for the small-amplitude free vibrations of a one-dimensional axially moving structure. The aim of the systematic presentation is to clearly expose the construction of the model, from physical principles through to the final linearized equations, which are then used to determine the stability of the physical system by linear stability analysis. We consider the construction of some commonly used linear material models, including the linear elastic solid, and two viscoelastic solids, the Kelvin-Voigt solid and the three-parameter solid. We highlight the connection between the beam and panel (plate undergoing cylindrical deformation) models. We derive the weak forms of the governing partial differential equations, and devote special attention to deriving appropriate boundary conditions for axially moving structures. With the help of a nondimensional parameterization, we identify scalings for which the physical system behaves identically. Finally, in the numerical examples, we present the results of a linear stability analysis for isotropic linear elastic and Kelvin-Voigt viscoelastic panels.