The natural compound eye has received much attention in recent years due to its remarkable properties, such as its large field of view (FOV), compact structure, and high sensitivity to moving objects. Many studies have been devoted to mimicking the imaging system of the natural compound eye. The paper gives a review of state-of-the-art artificial compound eye imaging systems. Firstly, we introduce the imaging principle of three types of natural compound eye. Then, we divide current artificial compound eye imaging systems into four categories according to the difference of structural composition. Readers can easily grasp methods to build an artificial compound eye imaging system from the perspective of structural composition. Moreover, we compare the imaging performance of state-of-the-art artificial compound eye imaging systems, which provides a reference for readers to design system parameters of an artificial compound eye imaging system. Next, we present the applications of the artificial compound eye imaging system including imaging with a large FOV, imaging with high resolution, object distance detection, medical imaging, egomotion estimation, and navigation. Finally, an outlook of the artificial compound eye imaging system is highlighted.