A curved vortex tube induces a self-induced velocity of the vortex in the direction of the centerline binormal vector. Different parts of the vortex tube will experience different self-induced velocity, leading to change in shape of the vortex. This chapter first reviews the geometry of space curves and its application to the vortex filament model. Both the Rosenhead and Crow cutoff approximations are introduced for regularization of the vortex filament equation. The local induction approximation is introduced, and its properties are examined. Theory for bending waves on a vortex tube is developed, both based on the small-amplitude Kelvin theory and the long-wavelength vortex filament theory. The Hasimoto solution for solitary twist waves on a vortex is discussed. The chapter lays the foundation for several different approximate methods for solution of bending motion of vortices, and it explores different types of vortex bending motions that are of particular interest to the understanding of vortex flows. Instability of an anti-parallel vortex pair is examined, including both the long-wave Crow instability and the short-wave elliptic instability. An application is examined for the problem of instability of a vortex ring, both in a quiescent fluid and in a straining or shearing flow.