Structures that are equipped with a periodic internal geometry to achieve a specific dynamical behavior, such as vibration isolation or nonreciprocal wave propagation are quite popular. Current designs include structures with periodic linear inserts, which are good at suppressing oscillations around a particular frequency. However, their bandwidth is rather low due to their linear behavior. We are exploring the use of geometric and parametric nonlinearities to create metastructures that can exhibit wider bandgaps and nonreciprocal behavior. These studies include systems under rotation and multi-frequency excitation, such as fluid-structure interactions.