The fields of computer vision and special effects production often require editing and synthesis of facial expressions, such as transfer one person's expression to another's face. It is crucial to synthesize realistic facial expressions to ensure accurate expression of a person's feelings, however, because expressions are very subtle and subject to individual differences, facial expressions generated using methods such as 3D facial reconstruction and face morphing are difficult to guarantee sufficient facial details and the workload is huge, requiring a lot of time and effort. Facial expression transfer techniques can synthesize a large library of face expressions while ensuring human privacy, and provide data support for related face research problems. With the development of deep learning, generative adversarial networks provide a new solution for facial expression transfer. The currently available GANs are CGAN, StarGAN, G2-GAN, GC-GAN, geometric feature and attribute tagging driven GAN, Expr-GAN, TER-GAN, and StyleGAN. This paper focuses on the above methods of facial expression transfer based on generative adversarial networks, and discusses their respective features and advantages; summarizes the available facial expression transfer; finally, based on the problems of existing methods, research directions for future work are proposed.