Bayer color filter array (CFA) images are captured by a single-chip image sensor covered with a Bayer CFA pattern which has been widely used in modern digital cameras. In the past two decades, many compression methods have been proposed to compress Bayer CFA images. These compression methods can be roughly divided into the compression-first-based (CF-based) scheme and the demosaicing-first-based (DF-based) scheme. However, in the literature, no review article for the two compression schemes and their compression performance is reported. In this article, the related CF-based and DF-based compression works are reviewed first. Then, the testing Bayer CFA images created from the Kodak, IMAX, screen content images, videos, and classical image datasets are compressed on the Joint Photographic Experts Group-2000 (JPEG-2000) and the newly released Versatile Video Coding (VVC) platform VTM-16.2. In terms of the commonly used objective quality, perceptual quality metrics, the perceptual effect, and the quality–bitrate tradeoff metric, the compression performance comparison of the CF-based compression methods, in particular the reversible color transform-based compression methods and the DF-based compression methods, is reported and discussed.