Natural visual systems have inspired scientists and engineers to mimic their intriguing features for the development of advanced photonic devices that can provide better solutions than conventional ones. Among various kinds of natural eyes, researchers have had intensive interest in mammal eyes and compound eyes due to their advantages in optical properties such as focal length tunability, high-resolution imaging, light intensity modulation, wide field of view, high light sensitivity, and efficient light management. A variety of different approaches in the broad field of science and technology have been tried and succeeded to duplicate the functions of natural eyes and develop bioinspired photonic devices for various applications. In this review, we present a comprehensive overview of bioinspired artificial eyes and photonic devices that mimic functions of natural eyes. After we briefly introduce visual systems in nature, we discuss optical components inspired by the mammal eyes, including tunable lenses actuated with different mechanisms, curved image sensors with low aberration, and light intensity modulators. Next, compound eye inspired photonic devices are presented, such as microlenses and micromirror arrays, imaging sensor arrays on curved surfaces, self-written waveguides with microlens arrays, and antireflective nanostructures (ARS). Subsequently, compound eyes with focal length tunability, photosensitivity enhancers, and polarization imaging sensors are described.