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Concept drift is one of the most frustrating challenges for learning-based security applications built on the close-world assumption of identical distribution between training and deployment. Anomaly detection, one of the most important tasks in security domains, is instead immune to the drift of abnormal behavior due to the training without any abnormal data (known as zero-positive), which however comes at the cost of more severe impacts when normality shifts. However, existing studies mainly focus on concept drift of abnormal behaviour and/or supervised learning, leaving the normality shift for zero-positive anomaly detection largely unexplored. In this work, we are the first to explore the normality shift for deep learning-based anomaly detection in security applications, and propose OWAD, a general framework to detect, explain and adapt to normality shift in practice. In particular, OWAD outperforms prior work by detecting shift in an unsupervised fashion, reducing the overhead of manual labeling, and providing better adaptation performance through distribution-level tackling. We demonstrate the effectiveness of OWAD through several realistic experiments on three security-related anomaly detection applications with long-term practical data. Results show that OWAD can provide better adaptation performance of normality shift with less labeling overhead. We provide case studies to analyze the normality shift and provide operational recommendations for security applications. We also conduct an initial real-world deployment on a SCADA security system.