Axial motion within a vortex core is important for regulating the variation in vortex core area along a vortex tube, and it is also responsible for certain vortex flow instabilities. Axial core flow can arise from several different mechanisms, including pumping of fluid into the vortex core within a vortex boundary layer, straining motion along the vortex axis, vortex sheet roll-up, or viscous decay of a vortex. Solutions for a strained vortex are introduced, including the Burgers vortex solution and Lundgren's strained spiral vortex solution. Solutions for axial motion generated by viscous decay and vortex sheet roll-up are introduced. Exact solutions for a vortex end wall region are introduced from Serrin and Long. The plug-flow model for transport of axial velocity in a vortex tube is developed. Solutions for area-varying waves on a vortex core are developed using both the small-amplitude Kelvin theory and the long-wavelength plug flow theory. The Bragg-Hawthorne equation is developed and used to derive the area-varying solitary wave solution. A series of problems are discussed describing instability criteria for vortex flows, jet flows, and swirling jet flows.