With extensive face images being shared on social media, there has been a notable escalation in privacy concerns. In this paper, we propose AdvCloak, an innovative framework for privacy protection using generative models. AdvCloak is designed to automatically customize class-wise adversarial masks that can maintain superior image-level naturalness while providing enhanced feature-level generalization ability. Specifically, AdvCloak sequentially optimizes the generative adversarial networks by employing a two-stage training strategy. This strategy initially focuses on adapting the masks to the unique individual faces via image-specific training and then enhances their feature-level generalization ability to diverse facial variations of individuals via person-specific training. To fully utilize the limited training data, we combine AdvCloak with several general geometric modeling methods, to better describe the feature subspace of source identities. Extensive quantitative and qualitative evaluations on both common and celebrity datasets demonstrate that AdvCloak outperforms existing state-of-the-art methods in terms of efficiency and effectiveness.