In this book, we discuss a variety of problems involving analyses of stability in mechanics, focusing especially on the stability of axially moving materials. This is a special topic encountered, for example, in process industry applications, such as in papermaking. In theoretical terms, the field of axially moving materials is located halfway between classical solid mechanics and fluid mechanics. The object of interest is a solid, but it flows through the domain of interest such as a particular section of a paper machine, motivating the use of an Eulerian viewpoint. In vibration problems in mechanics, a loss of stability is often (but not always) accompanied by a bifurcation in the complex eigenvalue curves describing the behavior of the system under the action of a quasistatically increasing external load. Thus in many of the chapters of this book, we will consider bifurcations in some form.