In the “Beyond Moore” era, the information device is expected to exhibit advantages including small sizes, high processing speed, and low power and dissipation. The novel magnetic information device with these advantages is made of heavy metal(HM)/ferromagnet (FM) composite. Owing to the asymmetric structure, the anisotropic exchange coupling named the interfacial Dzyaloshinskii–Moriya interaction (iDMI) is generated at the HM/FM interface. This iDMI influences the magnetic dynamics including ferromagnetic resonance (FMR), spin wave, and the motion of chiral DWs. These magnetic dynamic behaviors are the bases of the functions of novel magnetic information devices. Therefore, the influence of iDMI on the magnetic dynamics has attracted wide attention in recent years. In this topical review, we give a detailed introduction and discussion about recent investigation on the iDMI-relevant magnetic dynamics of the HM/FM bilayer system. This review consists of five sections: (1). the introduction about the background, the basic theory of magnetic dynamics and DMI; (2). the review about the effect of iDMI on the propagation of spin wave. Owing to the iDMI, the dispersion relationship of spin wave is asymmetric. This not only offers a precise method for measuring the iDMI constant, but also gives rise to potential application for novel magnonic devices. (3). the review about the effect of iDMI on the FMR. Unique iDMI-relevant mode was observed in the FMR spectra owing to the nonparallel alignment of magnetic moments. (4). the review about the motion of DWs with chiral structure due to iDMI. The iDMI plays a fundamental role in the high velocity of the chiral DWs. Meanwhile, the iDMI results in the tilting of DW plane, and the mechanism has been widely investigated. The tilting of the DW plane may be depressed by the interlayer exchange coupling. (5). finally, we summarize the review and give an outlook.