The application of holographic technology has enabled three-dimensional (3D) displays to develop rapidly, providing users an unparalleled visual experience. The calculation and generation processes of holographic images and video generate a considerable amount of data, causing great difficulties for transmission and storage. To meet the high-definition and real-time requirements of 3D displays, the compression of holographic data is critical. Since holographic images have distinct characteristics than ordinary images, it is difficult to achieve optimal results using existing compression techniques. This study introduces the main technical challenges of holographic compression and summarizes the commonly used metrics for holographic data compression. The cutting-edge technologies proposed by national research groups related to holographic-image-coding-quantization processing, transform-coefficient simplification, optimization of the international standard hologram format, and development of a new standard framework are discussed in detail, and their respective advantages and disadvantages are detailed. Finally, a future direction for research on holographic-compression technology is proposed.