The aeroelastic stability of simply supported, circular cylindrical shells in supersonic flow is investigated. Nonlinearities caused by large-amplitude shell motion are considered by using the Donnell nonlinear shallow-shell theory, and the effect of viscous structural damping is taken into account. Two different in-plane constraints are applied to the shell edges: zero axial force and zero axial displacement; the other boundary conditions are those for simply supported shells. Linear piston theory is applied to describe the fluid-structure interaction by using two different formulations, taking into account or neglecting the curvature correction term. The system is discretized by Galerkin projections and is investigated by using a model involving seven degrees of freedom, allowing for traveling-wave flutter of the shell and shell axisymmetric contraction. Results show that the system loses stability by standing-wave flutter through supercritical bifurcation; however, traveling-wave flutter appears with a very small increment of the freestream static pressure that is used as the bifurcation parameter. A very good agreement between theoretical and existing experimental data has been found for flutter amplitudes. The influence of internal static pressure has also been studied.