Modern cameras typically use an array of millions of detector pixels to capture images. By contrast, single-pixel cameras use a sequence of mask patterns to filter the scene along with the corresponding measurements of the transmitted intensity which is recorded using a single-pixel detector. This review considers the development of single-pixel cameras from the seminal work of Duarte et al. up to the present state of the art. We cover the variety of hardware configurations, design of mask patterns and the associated reconstruction algorithms, many of which relate to the field of compressed sensing and, more recently, machine learning. Overall, single-pixel cameras lend themselves to imaging at non-visible wavelengths and with precise timing or depth resolution. We discuss the suitability of single-pixel cameras for different application areas, including infrared imaging and 3D situation awareness for autonomous vehicles.