In this study, we consider a class of nonlinear aeroelastic stability problems, where geometric nonlinearities arising from large deflections and rotations in the structure interact with aerodynamic nonlinearities caused by moving shocks. Examples include transonic panel flutter and flutter of transonic wings of high aspect ratio, where the presence of both structural and aerodynamic nonlinearities can have a dramatic qualitative as well as quantitative effect on the flutter behavior. Both cases represent inherently nonlinear fluid-structure problems, where neglecting either the structural or the fluid nonlinearities can lead to completely erroneous stability predictions. The results presented in this paper illustrate the rich and in some cases surprising flutter behaviors of transonic wings, and the inherent limitations of the von Karman nonlinear plate model in strongly nonlinear fluid-structure interaction problems of this type. (c) 2008 Published by Elsevier Ltd.