The attenuation of low frequency elastic and acoustic waves remains a fundamental engineering challenge. Metamaterials and phononic materials have recently emerged as an attractive solution to this problem. Metamaterials are engineered materials that derive their effective properties from their sub-wavelength structure rather than material composition. Localized resonance of their sub-structures causes wave attenuation bandgaps due to energy sequestering and prohibits the propagation of incident waves. These local resonance bandgaps are distinct from those observed in phononic materials where attenuation bandgaps occur due to multiple scattering and interference effects between periodic elements. In this talk, I describe an application of local resonance and phononic bandgaps towards improving the dynamic behavior of composite sandwich structures. The high stiffness to weight ratio and the possibility of tailoring material properties according to specific needs make sandwich structures an ideal solution for high performance applications. However, poor performance under dynamic loading is a major roadblock in the widespread adoption of such structures. I will demonstrate that periodically embedding resonant substructures into the sandwich core drastically improves their dynamic performance without a significant weight penalty. I will also discuss the possibility of obtaining super-wide wave attenuation bandgaps at low frequencies through the interaction of phononic and local resonant bandgaps. These results are the first steps towards the development of lightweight sandwich composites capable of high performance under dynamic environments.