Abstract The present thesis investigates the dynamics of wave structures in dense planetary rings by employing hydrodynamic models, along with local N-body simulations of the particulate ring flow. The focus is on the large-scale satellite induced spiral density waves as well as on the free short-scale waves generated by the viscous overstability in Saturn's A and B rings. An analytic weakly nonlinear model is derived by using perturbation theory based on multi-scale methods to compute the damping behavior of nonlinear spiral density waves in a planetary ring subject to viscous overstability. In order to study the complex spatio-temporal evolution of these wave structures, numerical schemes are developed to integrate the hydrodynamical equations in time on large radial domains, taking into account collective self-gravity forces of the ring material, as well as the forcing by an external satellite. The required numerical stability and accuracy is achieved by applying Flux-Vector-Splitting methods aligned with advanced shock-capturing techniques. The free short-scale overstable waves are also investigated with local N-body simulations of the sheared ring flow. In particular, the influence of collective self-gravity between the ring particles as well as the periodic forcing due to a nearby Lindblad resonance on the overstable wave pattern is considered. The linear stability criterion for spiral density waves in Saturn’s rings is found to be identical to the condition for the onset of spontaneous viscous overstability in the limit of long wavelengths and agrees with the stability criterion for density waves derived by Borderies et al. within the streamline formalism. The derived nonlinear damping behavior of density waves can be very different from what has previously been thought. The role of collective self-gravity on the nonlinear evolution of short-scale overstable waves is determined, reconciling the partly contradicting results of previous studies. It is shown that collective self-gravity plays an important role in setting the length-scale on which the nonlinear overstable waves saturate in a planetary ring. A co-existence of spiral density waves and short-scale overstable waves is modeled in terms of one-dimensional large-scale hydrodynamical integrations. Due to the restriction to one space dimension, certain terms in the hydrodynamical equations that arise from the spiral shape of a density wave need to be approximated based on the weakly nonlinear model. These integrations reveal that density waves and spontaneous viscous overstability undergo complex interactions. In particular it is found that, depending on the relative magnitude of the two wave structures, the presence of short-scale overstable waves can lead to a damping of an overstable density wave and vice versa, density waves can suppress overstability. The effect of a density wave on the viscous overstability is also studied in terms of a simplified axisymmetric model of a ring perturbed by a nearby Lindblad resonance. A linear hydrodynamic stability analysis and local N-body simulations of this model system conform the corresponding results of the large-scale hydrodynamical integrations.