The study of vortex rings provides a first glimpse of some of the unique phenomena exhibited by three-dimensional vortices. We examine various processes leading to formation of vortex rings and summarize vorticity transport theory for the special case of axisymmetric flows. The Helmholtz solution for the flow induced by a thin-core vortex ring is developed, and an expression for the self-induced velocity of a thin-core vortex ring is derived. The Hill's spherical vortex solution is developed and the Norbury-Fraenkel family of vortex rings is discussed. Applications including the interesting phenomenon of leapfrogging of vortex rings and the generation of vortex rings by a falling cluster of particles are discussed. This chapter focuses on axisymmetric vortex ring flows, and fully three-dimensional problems (e.g., vortex ring instability) are examined in Chap. 7 .