Abstract. Edge illumination (EI) is an x-ray phase-contrast imaging technique, exploiting sensitivity to x-ray refraction to visualize features, which are often not detected by conventional absorption-based radiography. The method does not require a high degree of spatial coherence and is achromatic and, therefore, can be implemented with both synchrotron radiation and commercial x-ray tubes. Using different retrieval algorithms, information about an object's attenuation, refraction, and scattering properties can be obtained. In recent years, a theoretical framework has been developed that enables EI computed tomography (CT) and, hence, three-dimensional imaging. This review provides a summary of these advances, covering the development of different image acquisition schemes, retrieval approaches, and applications. These developments constitute an integral part in the transformation of EI CT into a widely spread imaging tool for use in a range of fields. © The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.