The chapter describes the various forms of periodic structures, i.e., sonic crystal, metamaterials, locally resonant structures, etc. Since the last few decades, periodic structures have been widely used by researchers as wave attenuators for particular frequencies. As the name suggests, periodic structures are those in which an object is embedded with a regular interval into another material. The shape of the object, type of periodicity, density ratio, periodic constant, etc., may vary according to the application. This chapter involves the various applications of periodic structures and acoustic panels when these are subjected to audible waves and elastic waves. The chapter also discusses the various numerical techniques, i.e., plane wave expansion method, the Webster horn method, finite element method, and analytical methods to compute the sound attenuation properties of periodic structures and panels. The chapter discusses the combined behavior of sonic crystal with acoustic panels. It includes various recent research findings and the future scope of periodic structures and panels.