Structural thin shells can be found in many engineering applications such as roofs, pipes, and tanks to name just a few of their applications. In some of their applications, the thin shells can have a structural interaction with, for example, an elastic base and/or external/internal fluid. Due to the slenderness of these structures and the characteristic of external excitation, they can exhibit large vibration amplitudes, requiring analysis about their structural stability. To describe the nonlinear dynamical behavior of these structures, the equilibrium equations are described by nonlinear partial differential and some numerical technique is applied to discretize these equations. Thus, this work presents some important aspects of nonlinear dynamics and structural stability of thin shells structures type. For that, the basic equilibrium equations for thin shells and the perturbation method, associated with Galerkin method, to discretize these equations are shown. Once the system of discretized equilibrium equations is obtained to a specific geometry of thin shell, the nonlinear dynamical behavior and the structural stability of the thin shells are investigated through the phase-portraits, bifurcation diagrams and basins of attraction. Special attention is given to nonlinear phenomena such as modal coupling, modal interaction, internal resonances, and influence of initial geometrical imperfections. From those obtained numerical results, complex scenarios in bifurcation diagrams, with important changes in structural stability; competition between different attractors, since this type of dynamical systems can present multiple potential-well; and changes in the bifurcation diagrams due to multiples internal resonances can display important role in the nonlinear dynamical behavior.