This paper treats rotationally symmetric motions of axisymmetric shells. It derives the governing equations in a convenient form and determines their mathematical structure. The complicated governing equations have the virtue of the far simpler equations for axisymmetric motions that there is but one independent spatial variable. Consequently the constitutive equations enjoy convenient monotonicity properties. The richness of rotationally symmetric motions is illustrated by a numerical treatment of an initial-boundary-value problem for a nonlinearly elastic cylindrical shell. This paper discusses the subtle question of nonexistence of general axisymmetric motions of axisymmetric shells. It briefly treats spatially autonomous motions, which are governed by ordinary differential equations in time.