Deepfakes are notorious for their unethical and malicious applications to achieve economic, political, and social reputation goals. Recent years have seen widespread facial forgery, which does not require technical skills. Since the development of generative adversarial networks (GANs) and diffusion models (DMs), deepfake generation has been moving toward better quality. Therefore, it is necessary to find an effective method to detect fake media. This contemporary survey provides a comprehensive overview of several typical facial forgery detection methods proposed from 2019 to 2023. We also analyze and group them into four categories in terms of their feature extraction methods and network architectures: traditional convolutional neural network (CNN)-based detection, CNN backbone with semi-supervised detection, transformer-based detection, and biological signal detection. Furthermore, it summarizes several representative deepfake detection datasets with their advantages and disadvantages. Finally, we evaluate the performance of these detection models with respect to different datasets by comparing their evaluating metrics. Across all experimental results on these state-of-the-art detection models, we find that the accuracy is largely degraded if we utilize cross-dataset evaluation. These results will provide a reference for further research to develop more reliable detection algorithms.