Holography has the potential to become the ultimate 3D experience. Nevertheless, in order to achieve practical working systems, major scientific and technological challenges have to be tackled. In particular, as digital holographic data represents a huge amount of information, the development of efficient compression techniques is a key component. This problem has gained significant attention by the research community during the last 10 years. Given that holograms have very different signal properties when compared to natural images and video sequences, existing compression techniques (e.g. JPEG or MPEG) remain suboptimal, calling for innovative compression solutions. In this paper, we will review and analyze past and on-going work for the compression of digital holographic data.