Book excerpt: This book deals with theoretical aspects of modelling the mechanical behaviour of manufacturing, processing, transportation or other systems in which the processed or supporting material is travelling through the system. Examples of such applications include paper making, transmission cables, band saws, printing presses, manufacturing of plastic films and sheets, and extrusion of aluminium foil, textiles and other materials. The work focuses on out-of-plane dynamics and stability analysis for isotropic and orthotropic travelling elastic and viscoelastic materials, with and without fluid-structure interaction, using analytical and semi-analytical approaches. Also topics such as fracturing and fatigue are discussed in the context of moving materials. The last part of the book deals with optimization problems involving physical constraints arising from the stability and fatigue analyses, including uncertainties in the parameters. The book is intended for researchers and specialists in the field, providing a view of the mechanics of axially moving materials. It can also be used as a textbook for advanced courses on this specific topic. Considering topics related to manufacturing and processing, the book can also be applied in industrial mathematics.