Limited by the planar imaging structure, the commercial camera needs to introduce additional optical elements to compensate for the curved focal plane to match the planar image sensor. This results in a complex and bulky structure. In contrast, biological eyes possess a simple and compact structure due to their curved imaging structure that can directly match with the curved focal plane. Inspired by the structures and functions of biological eyes, curved vision systems not only improve the image quality, but also offer a variety of advanced functions. Here, we review the recent advances in bioinspired vision systems with curved imaging structures. Specifically, we focus on their applications in implementing different functions of biological eyes, as well as the emerging curved neuromorphic imaging systems that incorporate bioinspired optical and neuromorphic processing technologies. In addition, the challenges and opportunities of bioinspired curved imaging systems are also discussed. Graphical abstract