Acoustic metamaterials are of growing interest due to their ability to manipulate the propagation of acoustic waves in an extraordinary manner to benefit various applications, such as communications, biosensing, and medical diagnosis and therapy. Among various construction methods of acoustic metamaterials, artificially engineered elastic periodic structures, known as phononic crystals (PCs), are the strongest candidates since they exhibit complete phononic band gaps and negative refractions due to the periodicity of the structure. In this book, a new class of acoustic metamaterials-gradient-index phononic crystal (GRIN PC)-is introduced to overcome the limitations of regular PCs and further enrich the control over acoustic waves. GRIN PCs with different gradient profiles are designed to guide acoustic waves in extraordinary manners that are not shown in nature or regular PCs. A practical method for tuning the phononic band gaps of a PC is investigated. The concepts presented in this book serve as important foundations for the future development of acoustic devices.