In this article, we review linear and nonlinear wave propagation in granular metamaterials with local resonances. The granular structures having inherent nonlinearity in contact interactions can support various wave dynamics from nearly linear to highly nonlinear regime depending on the pre-compression and wave amplitudes. Therefore it has been widely used for an ideal test-bed to realize different types of wave structures. The linear and nonlinear waves in granular metamaterials show distinctive features compared to those of conventional granular crystals due to local resonances. Moreover, the effect of local resonance on linear waves is substantially different from that on nonlinear waves in granular systems. For the comprehensive understanding of the effect of local resonances in granular metamaterials, their linear and nonlinear wave dynamics are compared with those of conventional granular crystals in both monoatomic and diatomic configurations.