The main goal of High Dynamic Range Imaging (HDRI) is precise reproduction of real world appearance in terms of intensity levels and color gamut at all stages of image and video processing from acquisition to display. In our work, we investigate the problem of lossy HDR image and video compression and provide a number of novel solutions, which are optimized for storage efficiency or backward compatibility with existing compression standards. To take advantage of HDR information even for traditional low-dynamic range displays, we design tone mapping algorithms, which adjust HDR contrast ranges in a scene to those available in typical display devices.