Natural compound eyes (NCEs) are the most abundant and successful eye designs in the animal kingdom. An NCE consists of a number of ommatidia, which are distributed along a curved surface to receive light. This curved feature is critical to the functions of NCE, and it ensures that different ommatidia point to slightly different directions and thus enables panoramic vision, depth perception, and efficient motion tracking while minimizing aberration. Consequently, biomimetic curved artificial compound eyes (BCACEs) have garnered substantial research attention in replicating the anatomical configuration of their natural counterparts by distributing ommatidia across a curved surface. The reported BCACEs could be briefly categorized into 2 groups: fixed focal lengths and tunable focal lengths. The former could be further subcategorized into simplified BCACEs, BCACEs with photodetector arrays within curved surfaces, and BCACEs with light guides. The latter encompasses other tuning techniques such as fluidic pressure modulation, thermal effects, and pH adjustments. This work starts with a simple classification of NCEs and then provides a comprehensive review of main parameters, operational mechanisms, recent advancements, fabrication methodologies, and potential applications of BCACEs. Finally, discussions are provided on future research and development. Compared with other available review articles on artificial compound eyes, our work is distinctive since we focus especially on the “curved” ones, which are difficult to fabricate but closely resemble the architecture and functions of NCEs, and could potentially revolutionize the imaging systems in surveillance, machine vision, and unmanned vehicles.