Reciprocity is a generic feature in various physical systems because it is directly associated to the symmetry of physical laws under time reversal. The breaking of the reciprocity is vitally important in numerous fields, such as the control of energy flux, imaging technologies and topological insulators. For instance, electrical diodes. as a typical nonreciprocal representative in electronic systems. have already contributed to substantial scientific revolutions in many aspects, including integrated circuits, lighting and signal indication. Motivated by the electrical diodes, thermal diodes exhibiting the rectifying effect on thermal energy are achieved in nonlinear lattices. Similarly, the nonreciprocity in photonic systems can be realized by many mechanisms, such as the nonlinearity of optical medium and the mode conversion. Synchronously, the concept of nonreciprocity further extends to phononic systems. The main means to achieve acoustic nonreciprocity are based on the nonlinearity of acoustic material, dynamic element, and magnetoelastic interaction. However, the nonlinear approach introduces the inherently low conversion efficiency, the dynamic scheme has difficulty in application at high-speed modulation or on nanoscale, and the method based on magnetoelastic interaction has small bandwidth and high frequency, which have severe limitations for practical application of the nonreciprocity. From the perspectives of principles, history, and problems, this paper reviews in detail the progress of the nonreciprocity based on nonlinearity, dynamic components, and magnetoelastic interactions. Some existing problems with these methods were pointed out. For example, the nonreciprocal solution based on nonlinear media has low energy conversion efficiency, it is difficult to miniaturize the nonreciprocal device based on dynamic components because of itself complexity, and the bandwidth of nonreciprocity based on magnetoelastic interaction is narrow. Correspondingly, improvement of the energy conversion efficiency based on the nonlinear medium, more reasonable design of a dynamic modulation system, and widening broadband of nonreciprocity in the magnetoelastic system, can be researched in the future. For example, in order to reduce the nonreciprocal frequency of magnetoelastic system, it is worth studying the unidirectional transmission of sound waves in the magnetoelastic system under a dynamic magnetic field. For the sake of realizing a nonreciprocal system with no external power source and high conversion efficiency, the nonreciprocity of acoustic waves in a magnetoelastic system under a zero magnetic field may become a focus of future research. In addition_ since each scheme has its advantages, the combined effects of multiple schemes may overcome the shortcomings of a single scheme and achieve more significant nonreciprocal effects. So far, the combined effects between nonlinearity and dynamic components have not received much attention. We expect that research of nonreciprocity will be extended to both nonlinear and spatiotemporally modulated systems in the future. The combined effects between nonreciprocity and magnetoelastic interaction may become an important content of future research. Finally, considering practical engineering application, there are still many aspects worthy of further study, such as the miniaturization of nonreciprocal devices, optimal design of nonreciprocity based on optimization theory, and adjustable nonreciprocity based on smart materials or structures. In short, the nonreciprocity of acoustic metamaterials is an important field of scientific research, which has potential applications, such as rectification, vibration and noise reduction, sensing, and bioimaging.
