Abstract: Disordered hyperuniform structures are a unique state of matter that exhibit significantly suppressed density fluctuations at large length scales similar to those found in crystals or quasicrystals yet without any long-range orientational order. In recent years, this exotic state has been widely observed in a variety of natural systems. Moreover, scientists have recognized that hyperuniformity is closely related to the fascinating mechanics of nonequilibrium statistical physics, which can also be realized in many nonequilibrium systems. Displaying coexistence of local structural randomness and large-scale fluctuation suppression, this structure holds great potential for applications in materials science, optics, and related fields, driving hyperuniformity to emerge as a novel interdisciplinary research frontier. This article reviews recent key developments in the field, with a focus on dynamic hyperuniform structures found in several representative nonequilibrium systems, including hyperuniform structure near critical points of absorbing phase transitions, nonequilibrium hyperuniform fluids, and hyperuniform structures formed in active matter and biological systems.