Over the past decades, photonics has transformed many areas in both fundamental research and practical applications. In particular, we can manipulate light in a desired and prescribed manner by rationally designed subwavelength structures. However, constructing complex photonic structures and devices is still a time-consuming process, even for experienced researchers. As a subset of artificial intelligence, artificial neural networks serve as one potential solution to bypass the complicated design process, enabling us to directly predict the optical responses of photonic structures or perform the inverse design with high efficiency and accuracy. In this review, we will introduce several commonly used neural networks and highlight their applications in the design process of various optical structures and devices, particularly those in recent experimental works. We will also comment on the future directions to inspire researchers from different disciplines to collectively advance this emerging research field.