Acoustic metamaterials, structured to produce anomalous reflection and refraction indices that are not found in conventional materials, are gaining prominence in engineering applications. These artificial structures have enabled novel functionalities, such as negative effective properties, extraordinary wave manipulation, enhanced sound absorption and insulation, cloaking, acoustic wave focusing, and efficient energy harvesting. To evaluate the research progress in the field of acoustic metamaterials, we take a novel viewpoint, tracing the development from passive acoustic metamaterials to active piezoelectric acoustic metamaterials. The article summarizes recent research progress in acoustic metamaterials, with the first part describing passive acoustic metamaterials and the second part moving on to active piezoelectric acoustic metamaterials and metasurfaces. The topics covered include their general definition, mechanisms, classification, structure, and potential applications. Finally, we survey the current technical challenges from a practical engineering standpoint and discuss the future outlook in this field.