Flat and uniform plate-like materials made of rubber, glass, and metal can be utilized as sound insulators. The weights of these materials need to be increased to increase the sound insulation effects according to the mass law. In contrast, acoustic metamaterials with periodic structures composed of numerous resonators can break the mass law limit at certain frequencies. However, their feasible applications are still limited because of the time-consuming processes for assembling and integrating resonators. Recently, our group reported an acoustic metamaterial based on a polymer sheet that mutually connects spring-mass local resonators. The integrated structure enables high-throughput fabrication through a polymer molding method and single-step implementation on target objects. It is expected that acoustic metamaterial sheets and the customization of the sheet designs according to individual requirements will offer practical and innovative solutions for various problems related to noise and vibration. In this Focus Review, the polymer-based material design, functional control based on the physical properties of polymers, and applications to sound insulators and vibration dampers are described. Moreover, useful insights of new acoustic materials that are developed by combining polymers and metamaterials are provided.