Artificial structures known as phononic crystals and acoustic metamaterials can be designed by spatially arranging one or more materials to obtain desired wave manipulation characteristics. The combination of various materials in complex composites is also a common feature of biological systems, which have been shaped in the course of evolution to achieve excellent properties in various requisites, both static and dynamic, thus suggesting that bioinspired concepts may present useful opportunities to design artificial systems with superior dynamic properties. In this work, a set of biological systems (nacre composites, spider webs, fractals, cochlear structures, and moth wings) and corresponding bioinspired metamaterials are presented, highlighting their main features and applications. Although the literature on some systems is vast (e.g., fractals), spanning multiple length scales for both structural and acoustic applications, much work remains to be explored concerning other biological structures (e.g., moth wings). Especially, bioinspired systems achieved by considering diverse objectives seem to be a promising yet relatively unexplored field of research.