The adverse effect of mechanical vibration is inevitable and can be observed in machine components either on the long- or short-term of machine life-span based on the severity of oscillation. This in turn motivates researchers to find solutions to the vibration and its harmful influences through developing and creating isolation structures. The isolation is of high importance in reducing and controlling the high-amplitude vibration. Over the years, porous materials have been explored for vibration damping and isolation. Due to the closed feature and the non-uniformity in the structure, the porous materials fail to predict the vibration energy absorption and the associated oscillation behavior, as well as other the mechanical properties. However, the advent of additive manufacturing technology opens more avenues for developing structures with a unique combination of open, uniform, and periodically distributed unit cells. These structures are called metamaterials, which are very useful in the real-life applications since they exhibit good competence for attenuating the oscillation waves and controlling the vibration behavior, along with offering good mechanical properties. This study provides a review of the fundamentals of vibration with an emphasis on the isolation structures, like the porous materials (PM) and mechanical metamaterials, specifically periodic cellular structures (PCS) or lattice cellular structure (LCS). An overview, modeling, mechanical properties, and vibration methods of each material are discussed. In this regard, thorough explanation for damping enhancement using metamaterials is provided. Besides, the paper presents separate sections to shed the light on single and 3D bandgap structures. This study also highlights the advantage of metamaterials over the porous ones, thereby showing the future of using the metamaterials as isolators. In addition, theoretical works and other aspects of metamaterials are illustrated. To this end, remarks are explained and farther studies are proposed for researchers as future investigations in the vibration field to cover the weaknesses and gaps left in the literature.